I bought a cheap trumpet and came with a mouthpiece that I've read it is sometimes recommended to change for another one. If I buy any mouthpiece of a different brand in the future , will it fit and work fine on my trumpet? Are receivers universal or should I be cautious when choosing one and look for a specific brand, characteristincs, or should I buy only the same brand of mouthpiece than my trumpet?
The best choice will be that you take your instrument with you to the shop when you once will change the mouthpiece. With lower instruments the size of the mouthpiece may always be a problem but
for trumpet mouthpieces, a "standard" crystallized for once. Nowadays a cone ratio of 1:20 is chosen for the cone for TRUMPET MOUTHPIECES. This is the case since about the 50s, so today almost all commercially available mouthpieces fit into all younger trumpets.
As has been said already, the taper on a trumpet m/p is an industry standard - so it's compatible with every trumpet m/p receiver. Some more esoteric trumpets have a one-piece mouthpiece/mouthpipe combination - but they are well out of the student price range.
Nearly all student model trumpets ship with a mouthpiece which is the equivalent of a Bach 7c. Yamaha use a different numbering scheme. You can tell it's a 7c if you blow gently up the tapered end and the m/p will whistle. No other m/p I know of will do that.
The 7c is OK - but most experienced players opt for a larger cup than a 7c. Bigger cup gives you a bigger sound, works a bigger portion of lip and needs more effort, so it's more physically tiring to play. Small m/p makes a bright sound, but can also make it harder to play loud low notes - depending on how thick your lips are...
Do beware the "new golf clubs" syndrome with mouthpieces - every vendor will be more than happy to sell you a m/p that promises better sound, higher notes and so on. But remember there's no quick fix. Only directed practice and a good embouchure will improve how you sound.
For this reason, it's better to live with a m/p for a while rather than buying one after playing it for only a few minutes in the shop. The best approach is to commit to a particular m/p even though it will be hard work. I've moved up gradually in cup size from a 7c as a beginner to a 1 now. If I'd been looking for a quick fix I'd have ditched the #1 as it was really hard tiring work to start with. But the investment pays off in time.
As others have noted: for trumpets, you can assume it will fit. However, it is still recommended to take your trumpet to the store. You will want to try out different mouthpieces to find one that is comfortable to you, and the best way to try them out is using your own instrument.
Remember: a mouthpiece is like a shoe; there is no universal "best fit", and you will have to try several until you find a good one