Welcome!

Tunnig issues can be from a variety of causes. SG's arn't known for rock solid tuning - though you should be able to improve things.

Firstly, I'd reccomend taking it to someone who has experience with working on guitars if you're not sure about something. I'd get them to go through the steps to re-string with me to see if I can improve how I do it. There are some tricks to re-stringing which can vary from guitar to guitar.

This isn't a direct improvement to tuning stability but an example of a trick I learnt when wathcing a friend restring SG style tune-o-matic bridges.

https://www.guitarworld.com/features/top-wrapping-les-paul

From the sound of it you may have a sticky nut. Quickest way to solve this is with a bit of graphite powder (pencil lead works well). If that's a bit messy, check out graphtech/tusq nuts, they may make a direct replacement.

Ultimately, guitars WILL go out of tune with playing, aging strings and seasonal climate changes. That's the nature of wood. That's why necks have truss rods

I had a Fender strat with all the tuning upgrades. Locking tuners, graphtech nut & saddles, locked out the tremelo, elixir strings, etc... And it still needed regular tuning. Honestly, I don't think it stayed in tune better than before the upgrades - BUT it was a lot easier & quicker to re-string and re-tune. Eventually I had to accept that it had a skinnier neck and this neck was more affected by climate changes (warm & cold days) than my other guitars.

Some guitars have composite (laminated wood) necks to help address this.