Help me to build my first stereo system

If it was me, I would definitely look at used gear, you are probably thinking the same thing. I would try to find a good Sony ES integrated amp or receiver, a Rega TT and some Monitor Audio bookshelf speakers. I have not searched what is currently available but I would think you could find those pieces close to your budget. If you wanted to have some extra fun you could try an AR XA turntable (and learn how to update it) and some EPI speakers (and learn how to recap the crossover). You would get a lot of help on the forums.

Good luck and have fun!

Off the wall answer: Get a good sound bar for your TV until you get enough budget. Edit: Or start anywhere and enjoy, you'll know where to go after that.

I was under similar budget a little while back and ended up getting a Rega Brio integrated for $300, Tekton Lore speakers for $500,  Schiit Audio dac for $200 running through an older NAD dvd/cd player and I’m extremely pleased in the result  for money spent.   You mentioned a turn table and I did not go that route.   However I think you can certainly put together a nice set up within you budget with some research and advice from the experts on the forum.

@augjonas Give the stuff a try! And of course -- remember that speaker position is what’ll make the magic happen. Away from the back wall. Up on stands. The two speakers and you in a triangle.

Find a good local hi-fi shop, of the independent, "high-end" variety. Even if you have to travel several hundred miles. Make it an overnight trip. If you provide your approximate location, perhaps those in your vacinity can suggest a dealer or two.

You can't expect perfect (and imperfect ;-) strangers to know what you find appealing in hi-fi gear. Helping you find a musically satisfying system is the job of a professional retailer. His profit margin is built into the price of every piece of hi-fi gear on the market.