technique

Getting Started with Illy
The first place to start is with Illy fine grind. That is what I use every day. It is consistent and decent in quality. If you are unable to pull a good shot with Illy than something is off. When you experiment, use it as a base point.

Grind is Everything
The single hardest part about making coffee with a Pavoni is getting the grind right. The first step is acquiring and getting to know a high-quality burr grinder.
Please, abandon any thoughts of using a common blade grinder (they have a little blender like blade in the center) as those little puppies wind up producing an inconsistent grind.
Pavoni makes a few grinders. One is a plastic unit available at Williams-Sonoma for around $50. These have a rather underpowered burr mechanism and are best avoided. I had one for 8 months and it did not do a very good job grinding espresso evenly. Eventually something inside the machine broke and it would only grind beans very coarsely.
I eventually purchased a Pavoni burr grinder that looked roughly as follows:
 

 
This unit matched the Pavoni machine series—albeit in chromed plastic, not chromed metal—and did a good job of grinding coffee evenly. With the front hopper removed, it will dispense coffee straight into your portafilter, although it will make something of a mess in doing so. In any event, when you remove the hopper, it will make a mess on your counter
Factory calibration of these units has been reported to be a problem. See David Bayer's solution. Try your machine first though. Mine worked well when set one notch to the right of #3. Eventually, like the first grinder, it broke.

Again, the Pavoni is very sensitive to grind. The importance of achieving a proper grind cannot be overstated. With the right grind, taste and crema magically appear. With the wrong grind, they are impossible to achieve.
Besides Illy, another option is to have your coffee ground at the place you buy it. Keep the coffee in a tightly sealed container, but don't put it in the freezer or in the fridge. Your coffee will last longer, but it will also be at the wrong temperature for brewing.
Getting your coffee ground at a store, however, is not easy. Most stores do not sharpen their blades properly and, in any event, do own machines with the kind of continuous fineness adjustment that the Pavoni grinder has.
Prior to buying my grinder, I tried perhaps a dozen espresso shops in search of the perfect grind. To be honest, I only found one: A New Leaf in Lake Placid New York, which I visited many years ago.

Roast
If you are interested in making truly good quality coffee, stick with good beans. Don't buy the supermarket brand. Avoid Starbucks and French Roast like the plague.These beans are over-roasted to ensure that quality variations, and with them, flavor, aren't discernible.

If you still worship at the altar of Starbucks, try a trip to Italy to strip away your delusions. Italian espresso uses a relatively light bean by American standards. Try a relatively light Yemeni or Ethiopian for example. For a time I'm used Ethiopian Harrar from "Mercato," a store in Los Angeles's Little Ethiopia. It's a good value.
Eventually I discovered that Trader Joe's Organic Espresso beans were pretty good. I used those for a time, but have defaulted to Illy since.

Quantity
The following is very important.
Forget about the visual statement that Pavoni is trying to make with the two spouts on the bottom of the portafilter.

There is no way to make two decent cups of espresso out of the amount of coffee that fits in the portafilter. One half cup is more like it.
If you doubt me, go to your nearest commercial espresso machine and take a look at the size of the basket it has. Notice that it's about twice the size of the one in the Pavoni. Cafes are not, in general, in the business of wasting coffee, so you don't have to think long about why the filter is twice the size.
The Pavoni's biggest flaw is that even the two-shot insert in the portafilter is under-sized for the average espresso shot. At best, it gives you a little bit more coffee than a ristretto.
Put in your coffee, filling it to the rim and tamp down. I use a scoop/tamper from Starbucks for this. I'd say that's about two tablespoons worth, but I'm too lazy to do the conversion. I do have one good thing to say about Starbucks: they sell excellent metal spoon/tamper contraptions for under $10. The tamper fits the Pavoni portafilter perfectly. Give it a try.

Brewing

Put the portafilter in and turn your machine on (red button on, white button on II). It's best to load your coffee in first so that the portafilter heats up as the machine heats up. You don't want your grounds to be shocked by the very hot water that will go through them. Make sure the valve for steaming milk is off. Don't let the amount of water in the glass sight get below 1/4". Running out of water will destroy your heating element.
Wait until the machine begins to sing through the pressure relief valve. When it is at full steam, the pressure is too great to make coffee properly. At this point, switch the white button to I. Once the pressure relief valve settles down and barely lets any steam escape, the machine is at the right pressure for making espresso.
Pull the lever up half-way a couple of times to warm the grounds. Then pull it up all the way, pause a second as the water gets drawn into the chamber and pull down slowly. If you have to use lots of force, your grind is too fine or you have tamped too hard. This will wear out your machine's parts prematurely. If there's no resistance whatsoever, your grind is too coarse or there is too much steam pressure. The latter only be the case if the relief valve is singing.
If your grind is set correctly, the coffee that should come out through the entire downwards stroke should be relatively dark brown. If the flow gets appreciably lighter, then pull the cup out from under the spout.
If you are unhappy and frustrated for some reason at this point, try it a few times. Pavonis can be inconsistent. Maybe its that tamping is so variable, I'm not sure why. if your problem continues, it is probably because you don't have a good grind. This is not your Pavoni's fault.

After you've finished making the espresso, go ahead and steam the milk if you want. Make the froth first by keeping the tip of the wand just under the top of the milk then lower the wand into the milk to heat it.
Don't forget to keep your Pavoni clean. If you do visit an espresso bar in Italy, you'll notice that the barista spends his idle moments cleaning and polishing his machine as if it was a prized Ferrari. Hey, you bought a Pavoni, so you obviously are of the same mentality. Keeping it clean will prolong its life.

Crema and Cleanliness
David Bayer :
"Crema is incredibly sensitive to the cleanliness of the basket area, like beer heads are sensitive to detergent residue in beer mugs. No amount of cleanliness is overdoing it; crema is a fragile creature which responds to total cleanliness. I use bottled or filtered water, and occasionally I surprise myself by swapping in an empty basket at the last minute, to see if I would want to drink the water that comes out.
When the machine is cool, I often hold it upside down under a powerful hot faucet (with practice the water stays in the general vicinity) then swab the inside rim with both ends of a wet Q-tip, then pump out the accumulated water. If I'm up for it, I then run a tank of boiling water through, taking care not to run it dry.

Before and between cups I blast a bit of plain boiling water through the empty basket holder to heat and clean everything. With a high enough catch cup this doesn't get water everywhere. Electric pump owners could get scalded trying the same thing. Friends rely on Q-tips and a wet towel, as an intermediate course."

Tamping
David Bayer again :
Filling the basket on its own leads to some optimizations and choices. I overfill with the basket on a catch plate, using a knife with blade held *vertically* to trim the fill to the brim of the basket. Because my grounds have fallen loosely from a hand grinder like sifted flour, this accurately controls the fill. I now tap the basket against the plate a number of times, which (I believe) creates a more interlocked crystal structure between the bean particles, lessening the risk of the water forcing a crevice shortcut all the way through. When one instead tamps from above, one differentially compresses the upper beans more (think of a 9 car pileup), which I can't see as desireable. I tamp a bit at the end to "polish" the surface, but I don't understand my motivation in doing so. I suspect that this smooths the effect of further compression as one installs the basket holder.

By leaving the last basket in place to drip for a few hours before removing, I can do a visual autopsy on how the tamping went. The contents should have drip-dried into a satisfyingly brick, not crumbly with mud still on top.
All this is critical if one is teasing the edge of the envelope on grinding as finely as viable, chasing the age of the roast. By grinding a bit coarser, the whole process gets much more reliable, but as long as I've got more coffee in the house I'm always willing to view the Pavoni as a 'one-armed bandit,' hoping to win the best cup of espresso I've ever had.

Pressure and Safety
David Bayer :

New machines have tightest seals, so they are most prone to brown spray coming out of the gruppo after the basket holder is released. Pressure is trapped in the lever-basket assembly just after pulling an espresso. Wait a bit, and slowly turn the basket holder to the first angle at which any pressure starts to release, and leave it there for a few moments. If there is no pressure in the main tank (i.e. you are refilling it), then lifting the lever back up also works.
This issue arises in the most embarrassing moments, like when a party full of people want espressos, and get you rushing. The brown gush is just a reminder never to let any piece of machinery know you are in a hurry.
Pressure is dangerous, the temperature alone is safe if you don't touch it. I burned myself once figuring this out. Turn off the Pavoni, and open the steam wand into a receptacle. When it stops making any noise, lift the lever all the way up (to equalize pressure throughout, like popping your ears), then slowly twist off the top cap. If it makes absolutely any noise, stop and wait a bit more. I got away a few times with just turning the cap to this point, but I turned a bit too far once, which is how I scalded myself.
Once the cap is off, refill till the level is almost out of view in the glass tube, retighten, close the steam wand, and fire up. Reheating takes much less time than a cold start. Some people leave the wand open; I open it after reheating to clear trapped water, but leave it on for the reheat like a lid on a pot I want to boil.

Timing
David Jenkins:
I've had my Pavoni Europiccola in daily use since 1977. I have installed a Pavoni Professional pressure gauge on the machine so that I can get consistent temperature. For the benefit of all you that do not have this, I measured the TIME it takes to get to the proper brewing temperature; it's been pretty consistent: Fill the machine half full of water. Open the steam wand valve. Turn both switches ON. When the steam wand starts to sputter, give it five seconds to blow the air out from above the water and the steaming to begin. Then close the valve, wait SIXTY SECONDS and turn the "HI" switch off. This is a good brewing temperature (around 192 degrees F). Make your coffee, THEN turn the "HI" switch back on and bring it up to the temperature where the safety valve hisses a bit to steam your milk. My biggest mistake over the years was steaming the milk first. If your water is at that high a temperature, you'll "burn" the coffee and it will be bitter. Try the technique above, it made a dramatic improvement in my coffee. A good information source is Espresso from Bean to Cup by Nick Jurich, available from Thanksgiving Coffee at 1-800-648-6491 for about twelve dollars. It gives detailed instructions on grinding, tamping, frothing milk, temperatures and pressures, and also has a very good "branching" troubleshooting guide (if you get result C, use a finer grind; if you get result B, use more coffee, etc.). It helped me a lot.

Robert Stokes has another suggestion:
He gets good results by grinding to gritty, rather than powdery, and tamping hard. He says he's also done well with extremely fine (pre-ground) coffee and light tamping, but feels he can perform a hard tamp more consistently than a light one.