Placing all your cards into a sleeve isn't a bad idea, just make sure you have enough sleeves for it! Usually the really cheap penny sleeves (like
these ones!) are easy and economical to get lots of, so when I've worked in card shops we'd always have everything in those.
My go-to suggestion would be to put anything holographic or foil into a sleeve like you said you're doing. I think that's a great idea, since it's not always easy to tell exactly how much a card could be worth just based on how shiny it is. Asking any local card shops or checking online could be worthwhile if you want to use high quality sleeves for specific cards.
I think if regular, cheaper cards are stored in a cardboard box they should be totally okay, as long as that box isn't at risk of something like having the cards spill out, or being on the floor where liquid could spill on them.
When I have specific cards I want to take really good care of, I'll double-sleeve them (with outer sleeves like
these ones) and put them in my binder, but for my cheapest cards I'm okay with simply sorting them and placing them in a simple card box like
this one, without sleeves. Sometimes it can really just depend on how big your collection is, how much you care about the cheaper cards, and how much effort you're willing to maintain them.
Hope this helps, and hope your cards stay safe!
