It is tough to break away from active speakers on a budget, as the whole chain has to be replaced. Staying within your budget, here is a system that will be a big upgrade over your current setup.
I have only heard ELAC speakers at an audio show, but I was always impressed with the sound, a great value. There are several models that will work for you, these have a front port which gives you some placement options.
https://www.crutchfield.com/p_970DB62/ELAC-Debut-2-0-B6-2.html?tp=186
An integrated amp will give you the best bang for your buck. I restarted my audio journey with NAD, and it served me well. There are other options, but this is a pretty decent unit. It even has a phono amp built in which may be an improvement.
https://www.crutchfield.com/p_745C316V2/NAD-C316BEE-V2.html?cc=07
And you will need some speaker cables, Blue Jeans has a good product:
https://www.bluejeanscable.com/store/speaker/index.htm
These are all new products, you might be able go up the line with these brands by buying used. Check out hifishark for scouting used gear.