The most important thing to understand before you buy a new mouthpiece is that music is very subjective and one player will have far different results than another player using the exact same equipment. Much of this has to do with a players physicality. Other factors that change how you should choose your mouthpiece is your own personal ability.
What Do You Look For On a New Mouthpiece?
There are a few details about each mouthpiece to look for when selecting a new one. They are:
This image is from Bach's Accessory Catalog. It does a good job of illustrating basic design features for mouthpieces. In my opinion, it is a good place to start.
How Should You Choose a Mouthpiece?
First, when trying out new mouthpieces, make sure to always use your main instrument to try new mouthpieces. Mouthpieces will play differently in other instruments. It is best to try new mouthpieces in an instrument you are used to, mostly so you are familiar with the pitch tendencies of your instrument and the limits of your own instruments dynamic ranges.
When trying mouthpieces, play music that you are very familiar with. The goal is to see what mouthpiece is the most comfortable and what you sound best on. Try not to make a purchase based on the description of what the mouthpiece should sound like alone. Always try before you buy.
How to Select Mouthpieces for Beginners
If you are a very young player and Tuba is your first instrument, you might want to start with a very small cupped mouthpiece, so it is easier to make a sound. If you have experience playing another brass instrument, it may be easier for you to use a mouthpiece with a bigger cup. A local band director, who is also a Tuba player recommends that the students in his district start on a Conn Helleberg 7B.
Below are Bach's Tuba mouthpieces. The image is from Bach's Accessory Catalog. It is a good reference point to start from when selecting a mouthpiece for beginners.
Common Mouthpieces or Boutique Mouthpieces?
What is the truth? Is there a big difference between common mouthpieces made by major manufacturers and custom mouthpieces made by a boutique shop? Yes. Will every player immediately be able to tell the difference between the two? No. Here is why I think this is the case. A young player who doesn't have much experience isn't going to have the ability to discern the subtle differences between a common mouthpiece and a very expensive mouthpiece. Think of it like a student driver trying to maneuver a well tuned sports car. They might be able to make it stop and go, but they won't be able to use it to its full potential. This is also true of fine Tubas as well...
There is a great benefit to using finely made boutique mouthpieces, but if you are a beginner, you might be paying a lot of money but physically unable to reap the benefits of it.
In closing, don't get bogged down in details and numbers. Try many different mouthpieces and let your ears do the work. There are many useful charts and spreadsheets, but only playing a new mouthpiece will reveal how well of a fit it truly is. The real answer of how to choose a mouthpiece is try as many as you can, and pick what is best for you.
Listed below are some links to some mouthpiece comparison charts and potentially useful information.
Tuba Mouthpiece Chart on All Brass Radio.com
David Werden's Tuba Mouthpiece Comparison Chart
Woodwind and Brasswind Tuba Mouthpiece Comparison Chart
Musician's Friend - How to Select Cup Mouthpieces
PS - The picture at the top of this post is from: