Although there are a lot of great mouthpiece companies out there I suggest Schilke. The mpcs are arranged by size and that makes it easier in case you want to change the size or cup depth.
I would say start with a Schilke 12. It should cost you $ 30 in mailorder.
When to change mouthpieces is dictated by the sound and by flexibility. If you have trouble going from low d to low g as a lip slur then you might need more cup to let more lip vibrate. You want to increase in steps so that you don't overshoot where you need to be. That would have an adverse affect on endurance and range.
As for playing more than one mouthpiece. The easiest way is to always use the same rim and cup diameter. Even then you need to practice on each mouthpiece that you intend to play. If you change the rim or diameter then you need 4 or 5 times as much extra practice as you would if you didn't change the rim. There are several reasons why you might choose to play more than 1 cup depth / shape / backbore. These have to do with making playing easier.
With the proper practice you can learn to change your tone color . This can be done by slightly rolling out your lips, or making your lip aperture more rounded or even by adjusting your air column (resonance freq.). If you need more immediate results a deeper cup or even one with some v shape will help.
Mouthpieces come in sizes like shoes do. Lips come in different sizes as well. Besides lip size the strength of the embouchure also comes into play. Professionals play mouthpieces of ALL sizes.
Jake used to say that if your mouthpiece had a hole at each end, air could blow through and you sounded good then FORGET IT. "Mouthpieces big fat deal."