Learn Swedish: Gillar du kaffe? - Do you like coffee?

Swedish doesn't use a helper word like "do" to ask questions. Instead, you just put the verb first. The statement Du gillar kaffe (You like coffee) simply flips to Gillar du kaffe? (Like you coffee?) to become a question.

Could I use tycker om instead of gillar here?

Yes, absolutely! Tycker du om kaffe? means exactly the same thing. Gillar is just a single word and slightly more casual, making it very common in everyday spoken Swedish.

Why is there no article before kaffe?

Just like in English, when you are talking about a general uncountable substance like coffee, you leave out the article. You are asking if they like coffee in general, not a specific cup, so it is just kaffe rather than ett kaffe or kaffet.