Welcome to the home of CoffeeSnobs and a whole different world of great coffee [smiley=thumbsup.gif].
I guess its more about getting all the basics right on a consistent basis which is the best road to achieving consistently high quality shots of coffee. The Gaggia Classic is capable of pulling extremely high quality brews so long as the above receives the attention it deserves.
All that being said, the main issues are.... the Brew Water temperature must be maintained within the ideal range during the pour, nominally 93deg C. The Brew Water Pressure likewise, should be maintained within its ideal range, nominally 9-10 BAR.... and the shot duration within the 25-28 seconds ballpark. Regarding the pressure adjustment you mentioned.... this is in reference to the Pressure Limit Valve that resides in the downstream path from the Pump itself and is intended to protect the Pump from experiencing excessive pressures during the course of pulling a shot.
In some machines, this valve is easily adjustable and some others, its a fixed valve and Im not sure what type the Classic has fitted. In any case, if you were tempted to adjust this valve without having the appropriate pressure measuring equipment handy, then it would be just like shooting in the dark as to whether you end up with it adjusted properly. Im not a real fan of playing around with these unless you know what you are doing and have access to the appropriate tools and instruments. If the adjustment is carried out too far, then it is not beyond the bounds of reason that you may risk damaging the Pump or something else in the high pressure water circuit.
Anyway, if all of the aforementioned criteria are being met, then it comes down to the way you grind, dose, distribute and tamp your coffee prior to pulling the first shot. Theres lots of info around on what to do in this aspect of preparing for great coffee, including on this very forum, and this site is always worth a read too.... http://www.home-barista.com/.
What Ive discovered though, is that most of the gains in achieving really great coffee is to use high quality, freshly roasted beans ground just before filling the PF basket and pulling the shot. I havent been roasting my own coffee for all that long really, about 18 months or so but I have noticed that since I have been paying a lot more attention to achieving repeatable roast profiles for each particular bean variety, the quality in the cup has improved by leaps and bounds.
If you are not roasting your own beans yet, then it is definitely something you should give a try in the not too distant future..... its easy, almost falling off a log easy and cheap to get started. Failing being able to roast your own, you should always try and buy your beans from an artisan roaster in your vicinity who has a reputation for producing high quality roasts.
Well, thats about it in a nutshell.... hope there is something in there to help you out. All the best,