The trumpet can be quite a beast. Endless hours are spent practicing. Because of that, most trumpet players would like to see progress like this:

When in reality progress looks more like this:

Because the trumpet is such a beast to play, many players are looking for any sort of help they can get to increase their rate of improvement. Because it’s relatively inexpensive and easy to change, many players turn toward a new mouthpiece.
The True Role of the Mouthpiece
The shape and size of the mouthpiece change the character of the sound. A mouthpiece that is too small will give you a tone that is not your best. However a mouthpiece that is too big will often encourage you to work too hard. Trumpet players start to do weird things (unconsciously) to compensate for fatigue. So the best mouthpiece to play on, for most mortal earthlings, is in the middle of the road. Everything else should be considered specialized equipment.
Middle of the Road
By middle of the road, I mean a (Bach sizes:) 7C, 5C, or 3C (Schilke sizes: 11, 12, 13C4, respectively).
Bach Size Notation
In Bach size notation, the number is the cup diameter (higher number – smaller rim) and the letter is the cup volume (A is big, F is small).
Schilke Size Notation
Example Schilke size: 13C4c; 13 is the cup diameter (bigger number = bigger diameter), C is the cup volume (A=small, B=large), 4 is the rim contour (1=rounded, 5=flat), and c is the backbore (a=tight, e=large). Because standard sizes are omitted, this size would actually be written 13C4, because c is the standard backbore.
Specialized Equipment
The whole idea of trumpet equipment is this: you want to approach the instrument the same way all the time, but change your equipment to facilitate the appropriate tone quality. You want the right tool for the job, because otherwise you are going to end up creating a strange unnatural technique to make up for it. The 3C-7C is going to be fine for 90% of people doing 90% of playing. A very advanced high school student who happens to play lead in jazz ensemble might consider, under the guidance of a private instructor, a smaller mouthpiece to facilitate a bright tone to carry through the ensemble and a smaller cup volume to facilitate extended upper register playing. I play a Schilke 12A4a for lead and a Bach 7E for piccolo trumpet.
For 95% of playing I use a 7C. While I may be able to get slightly more overtones in the sound with a 3C, I am shooting for perfect technique and I don’t want to make a big compromise in technique for a small benefit in tone.
Also, I play on gold plated rims. Contrary to popular belief, it has nothing to do with aesthetics; I am allergic to nickel which is a common ingredient in silver plating. If you are not allergic to anything, save your money and get the usual silver plating. I also have a plastic mouthpiece which is especially nice for playing out in the cold.
Roadkill
It’s important to have an accurate aural model of a good trumpet tone in order to develop good playing technique. When I was younger, I listened to a lot of flugelhorn playing and really enjoyed the warm, mellow sound the instrument made. While you should not try to control your sound (you should discover your own natural tone using good technique), consciously or not, I began to imitate the flugelhorn sound with my trumpet playing. I ended up playing with my slide pushed all the way in and using the biggest mouthpieces I could find; a Bach 1”X” (larger cup volume than A) and after that a Monette B1-5D (often confused for a toilet bowl). This helped me get that warm, dark, flugelhorn-like tone I was looking for. It also led me to a tone very uncharacteristic of a trumpet and strange habits to compensate for my quick fatigue, as I was swimming in those mouthpieces! They would be appropriate for a professional orchestral player who knew how to use them, but not for me as a lowly high school student. I would have saved a lot of energy and frustration if I had just invested in a flugelhorn mouthpiece at the time to get that flugelhorn sound.
Roadkill Café
Sometimes younger students have old equipment that has survived from a relative’s days in the school band. A raw brass horn is ok, but be careful of a mouthpiece whose plating has worn down; it is very possible to get lead poisoning from an un-plated brass mouthpiece. Even if the equipment is well plated, if it is strange or otherwise unidentifiable, I always steer students to the middle of the road.
Goldilocks and the Three Trumpet Mouthpieces
Why are you still reading this? You should be practicing! I was just joking, but in reality, any time spent mulling over the question, “will this new fangled piece of equipment help me get better, faster?” is time better spent practicing.
No new piece of equipment is going to turn you into Maynard Ferguson or Wynton Marsalis.
Your equipment should help you and work for you; make sure you are not the one working for it.
If you are a mature player working in a special situation, then you need specialized equipment. If not, just get your 3C, 5C, or 7C and get to work practicing.
Listen to live, un-amplified trumpet performances regularly (even if it is just your instructor) to help you develop a strong, sound aural model of what a trumpet really sounds like, but otherwise let go and discover your true sound on the instrument.
If you really want to sound more like a flugelhorn, just get a flugelhorn already! Short of that, a flugel mouthpiece will dramatically alter your tone.
Ultimately, stay in the middle of the road and don’t let students play on anything unnatural. Good luck and happy practicing!