You may want to review this
thread on temperature management of the La Pavoni/Gaggia Factory to be able to get more than one or two good shots. A temperature strip on the group head really helps with this.
I find that I need to adjust my Gaggia Factory for a finer grind than my La Peppina or a Mypressi Twist in order to get a good shot with enough resistance to the pull. Standard rules should apply for dialing in a grinder, find the point where the shot chokes and back off from there. Dr. Gary suggested that you should use about the same pressure on the lever as you would when using an AeroPress, if that helps any. He helped me dial in and get started using the Factory.
I tend to use fresh home-roasted or commercial coffee and have heard that as coffee ages you need to grind finer and finer in order to get a good flow rate. The crema can also suffer on older beans. The Illy beans should be good the day that you open them, but may deteriorate faster then fresher, more local beans so you may need to tighten up the grind. The advice about freezing the beans once you open the can is sound, I do this when I buy a commercial espresso roast and generally keep an 8 ounce jelly canning jar out to use during the week for pulling shots with the rest in the freezer.