Learn Spanish (Latin America): Yo quiero comer algo, pela la manzana.

Pela is an affirmative tú command, which we use to tell someone we know well to do something. For regular -ar verbs like pelar (to peel), the informal command is made by taking the normal tú form (pelas) and dropping the -s.

How would this sentence change if I were speaking formally to someone using usted?

To give a command to someone using the formal usted, regular -ar verbs change their ending to -e. So, you would say pele la manzana instead of pela la manzana.

Why is comer left in its basic, unchanged form here?

When you use two verbs back-to-back, only the first verb changes to match the person. We change the first verb to quiero (I want), and leave the second verb in its basic dictionary form, comer (to eat).

Do I always need to say yo quiero, or can I just say quiero?

You can completely drop the yo and just say quiero. In Spanish, the -o ending on the verb already tells us that I am the one wanting. Adding yo here is optional and just gives a little bit of emphasis.