In general, it is a good idea to try your mouthpiece before you buy it, if you can. A good thing to know, however, is that it is very often cheaper to buy your mouthpiece on the internet. While you can't be guaranteed that your moupiece will suit you perfectly, for the price of 1 mouthpiece in a shop, you can often buy 2 on the internet! However, I also encourage the patronage of local music stores (especially if they provide a decent selection to choose from), and if not, the only way they will be able to be able to provide a decent mouthpiece stock is if they have a customer base to provide for.
Most people have at least some experience with the mouthpieces on the left. The problem is often when the player is no longer satisfied with the familiar equipment, this is good, it means that you are developing- but remember, hang on to your old familiar mouthpiece- it is an excellent reference point, and you may like to use it for a particular situation. The sound that you produce on your familiar mouthpiece will probably not alter that much with a new mouthpiece (unless of course you switch from a jazz to classical mouthpiece, or vice-versa) so, when you are looking for a new mouthpiece, try to pay more attention to things like good articulation, response and intonation- over the whole instrument- sound will come later. Also, prepare some reeds ("break them in" see: REEDS) and use many different reeds while testing a mouthpiece- don't use your familiar reed, this may work for your familiar mouthpiece, but remember, if you want to change mouthpiece, you might want to change reed too!
It might be a good idea to try different ligatures (or reeds) if you are simply looking to have "something more" to your sound. They are (sometimes) less expensive than a new mouthpiece, and if the mouthpiece works, then there's not really any point fixing what isn't broken.