I expect the Thargoids (eventually) and nothing else for a long time, if ever.
Any alien civilization that interacts with players primarily through combat will need very careful balancing. That's going to be complicated by the fact that the human ships will still be undergoing balance for their own ship-to-ship combat. It will be hugely disappointing if the Thargoids don't have their own unique weapons too. Thargoids running around with multicannons and pulse lasers won't cut it.
It's a massive enterprise to introduce a spacefaring, combat-capable alien civilization like this. So I don't expect more than the Thargoids for a long time.
I suppose FD could develop a few non-hostile spacefaring aliens, but that's still a lot of work with ship designs, trade interaction etc. And with guns available on every ship we fly, I'm not sure how well it would be accepted by players if we couldn't get in a scrap with them at some point.
That's the gameplay argument, anyway. In terms of "realism," I think there are some good arguments for a very low chance of encountering another civilization, at a similar enough level of technology, that we could interact with them in any meaningful way. The timescale of the Galaxy is too long -- i.e. advanced civilizations can rise and fall before you ever meet them -- and the mismatch between technology levels is likely to be too large. We can fight the Thargoids if they're somewhere close to us in technology, but any much higher advanced alien civilization would be a hopeless mismatch, even for basic communication. We don't try to talk to insects on our planet, after all.
That's the pessimist argument for extremely rare chance of meeting other alien civilizations, even given the huge number of stars in the Galaxy. The chances of finding a civilization at the "just right" Goldilocks stage to interact with us in a meaningful way, is very small.