Hello and welcome!
This is a pandora box and there is no one answer. I am beginner and I went through this already, sold my solid state amp and now selling my tube amp as well.
1/ If you go solid state or tube amp: just go straight to the home version of your dream amp, if any. For myself, that would be the brand new 5150 Iconic 15w but even that is too powerful with 1/4 reduction (6L6 power tube) so it would have to be used with studio monitors at home and that makes the price very high even before any pedals enter the equation. SO amp+pedals= exorbitant cost or you will never be “there.”
This route is expensive but you get a real amp if that is really your thing (turns out it ain’t mine).
2/ If you go modeller: I will probably get the Fractal FM 3 when I learn more. For now I use the 25$ Peavy Revalver 4 plugin software, which has fantastic sounds and a 100$ Boss Pocket GT which sucks in comparison but is very portable. The Headrush MX 5 is great at 40% the cost of the Fractal and you can get great patches from others for like 5-25$ each, like a Van Halen or Whitesnake patch in my case.
This is not cheap, that is not the point. FM is like 1000 and I just got 8" stereo monitors on sale for 300, normally they are 500, and that’s a good price. The point is quality + practicality, not price.
But for a start, you could get a cheap modeller and studio headphones.
3/ If you go modelling amp: this is the cheapest route by far and if you get a Blackstar COre you also get stereo. This is what I should have done at the very start but I talked myself into an Orange Crush and so it began.
This route is cheap as dirt. You can get a Blackstar ID COre for like 220 full price in the US whereas a Headrush MX5 with 5" speakers will run you about 750-800 if you pay full price.
And then you can decide which way to go.
In addition, I think the Fender Mustang Micro is fantastic add-on, ultra portable. I learned from my Boss Pocket GT but the fun only lasted a month. While tiny, it is extremely impractical and the Boss GT/GT1/GT10 modeller is nowhere close to the real thing on gain sounds. It has better high gain than gain but it is nowhere close to the Revalver 4, as they say it sounds “digital” whereas the Revalver 4 while it has quirks sounds great (the cleans are fine but I can play scales unplugged anyway, LOL).
In short, I would not try to improve on a limited amp cause eventually you will want more anyway and so instead of paying through the entire price scale from cheapest to priciest it is better to spend time and figure out what you really want with a cheap modelling amp or software or cheap modeller and get there asap.
Cheers!