What sub-field in linguistics should I study to help me learn foreign languages?

The answer to the headline question is very simple. None. Do what millions of successful language learners (and teachers) do, completely ignore linguistics. Just compare linguists and historians - you'll see that linguists are no better at learning other languages. Historically, advances in language learning have not generally come from advances in linguistic theory.

Perhaps the most impact linguistics has had on language education is functionalism. In that case, you will want to read Halliday and people writing in that tradition.

Personally, I found it useful to have a decent grounding in phonetics and phonology for language learning but only up to a point.

But it seems you're more interested in learning about languages.

If you're more interested in finding out about the variations in grammars across languages, you will be better served by contrastive linguists (or typology) like RMW Dixon - his Basic Linguistic Theory is a true delight. But the reading list would be practically endless. Bernard Comrie, William Croft, Martin Hasplemath, all of these people have lots of useful things to say.

If you're interested in how language works in general, then there are also many options. Even though I don't agree with his conclusions, I would still recommend Steven Pinker but there are many other popular authors. David Crystal is one that comes to mind.

Or simply, buy pretty much any introduction to linguistics and see what will strike you as worth exploring in more detail.