1. There are a lot of clubs in my school, like a sports club and an art club.

Question

2. There are a lot of clubs in my school, like the sports club and the art club.

(Do we have to use 'a' or 'the' in each sentence?)

Answer 1

Either one works just fine, depending on your meaning.

Using "a/an" means that you are referring to those clubs in general; they may or may not exist.

Using "the" means that you are referring to particular clubs that are already in existence.

Answer 2

Here is an interesting commentary. As for myself, I would have chosen such as. But both are correct.
http://www.usingenglish.com/forum/ask-teacher/13585-such-vs-like.html