Repotting tips?

I am pretty new to plants. I have five right now, I’ve had them for a year or two. They’re all still in their original black plastic pots.

I really want to repot them, for aesthetic and plant health reasons, but I repotted two plants in the past and I ruined them so bad.

The repotted plants were never able to stand upright again no matter how much soil I tried to pack in and they slowly died, diagonally. It was horrifying and I don’t want to mess up my current plants.