Well, the Yamaha student clarinets generally come with a Yamaha 4C mouthpiece. It really isn't terrible. However, if you think you hate your horn because of the mouthpiece, a) ask your instructor what he recommends and b) have him make sure you're either not doing something wrong or that there isn't something wrong with your horn, reed, mouthpiece, or ligature.
One of the reasons why I stopped playing clarinet for several years, when I was much younger, was because I didn't know I could get a different mouthpiece.
The best information that we can give you here is to make sure you have a decent mouthpiece. As I said, the 4C isn't bad.
We have addressed the topic in some depth, so no need to go over it again. As said, your instructor is the best person to talk to.
As far as a ligature is concerned, you just need one
that fits your mouthpiece and reed and isn't damaged in some way -- that also means that if it's supposed to have two thumbscrews, you have two thumbscrews. For me, a different ligature made some reeds either playable or playable longer than a stock sheet metal one, but it's not something to really worry about until you've played for a long while. Looks like a
standard Yamaha ligature is all of $10 on Amazon. If you want to go with more interesting looking, the
Luyben ones are about $11. Your local music store might be cheaper. FWIW, I generally used the sheet metal ones, a
Rovner, or a
Gigliotti. And I only used the sheet metal and Rovner ones because that's what my instructors used.
For more than enough ligature reading,
check out this thread from our clarinet content experts.