How to teach beginner ESL students the difference between ‘I like going to the cinema’ and ‘I’d like to go to the cinema’?

In short, you would not teach them the difference. Bringing up two very similar grammars, but quite with a different meaning, will just confuse people.

Generally, I introduce students to phrases using “I like…” quite early on. Especially young children like to say things like, “I like to eat pizza,” and, “I like going to the cinema.” It would be quite a long time later, possibly months or even years later, that we would get to grammar using, “I would…” and then later, “I would like to…”

I try to avoid contractions like “I’d” when teaching beginners. Once the student has learned “I would” then they can contract it. Many textbooks, written by Americans, use contractions all over the place form the very start, however, I was brought up British middle class and so rarely use in my personal speech.