OP, you have a very nice beginner system. Shoot, that is a worthy system of more than just beginners.
Because you have such a decent system, I would do what others say above and just go right to the speakers. I'd upgrade the loudspeakers first and then move to upgrading other things as you see fit.
I owned your integrated amp. Sure there are better ones out there but it is a fine integrated--feature rich, sounds very good, etc.
If you listen to tons of vinyl, you could start with just a dedicated phono preamp. I took that route when I was in nearly your exact spot--based on the advice of fellow member here. It was a wonderful upgrade. But at that time, my speakers weren't begging for an upgrade. I'm not saying yours are in necessarily in need of an upgrade...just if you are satisfied with them and you love vinyl you could get a big sonic improvement by going to a nice phono pre.
I agree with you too on the ELAC--I'd hold steady on that guy. Very nice product and no need to swap that out at all.
If you are considering speakers as the potential upgrade, can you tell us a bit about what you like, your listening preferences (how loud you listen, room size, genres, etc.?
Again, for what it's worth, I think you did a great job on that "beginner system". I bet it sounds pretty darn good.
Keep us posted! I love hearing about other folks' journeys.