Your teacher will tell you that the market is flooded with plastic mouthpieces which are really just made to make it true that a mouthpiece is 'included in the box'. They have a kind of brittle, ugly sound. Ebonite mouthpieces are more respectable and give a warmer tone. Metal mouthpieces are most often played these days. My advice is don't get hung up on mouthpieces until there is a modicum of embrouchure developed. The muscles need to develop in the lip and then there is a lifetime of brain wiring to get those nuances of vibrato etc.
I have a friend - Willie Garnett - a wonderful man in all departments generous kind and honest. He repairs horns in London, has done for forty or more years, looks after the top players, has had his own big band and a quintet, knows many of the greats. He is a true expert - not just someone with a certificate in something. He always tells me there is a lot of hype around horns and mouthpieces.
Assuming you cut out the real rubbish, so many horns blow well these days - if set up properly. BTW the way a horn can be checked is by using a cigarette paper - if it moves too easily under a closed pad then there is a problem. Another method is putting a light down the horn in a dark room. If light peeks through then the horn is leaking. A leaky horn will make a beginner (and pro) struggle hard. Pads: pads have a life, they are made of pigskin and get out of shape as the get moisture swelling them. Every few years of playing a horn needs repadding if played hard. Pad condition of a second had horn are frequently out, only an expert can really tell.
Back to mouthpieces - you can pay silly money, but I play a Otto Link New Yorker, which is I suppose a couple of hundred UK these days, this plays fine , I dont see the need ot pay £2000+.
Reeds, another factor anbd a very important one. A read has a life. Especially for beginners its a little harder to play when fresh, then gradually soften over the months, until discarded. Beginners play thinner reeds on the whole - size 2 ish. They do fx the tone. Stan Getz played a hard reed it was part of his tone - hard reeds sound softer, thin reeds sound brassier.

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