I recently changed the strings on my strat from 9s to 10s. I play with my fingers and not a pick, so lighter gauge strings for me are hard to pluck because they bend so easily when I pull on them, thus making it difficult to play fast runs or to tremolo pick. When I changed my strings, my trem started lifting, which is fine because I like to be able to go up and down in pitch. But now if I push down, the strings go down in pitch and don't go back up, when I pull up they go up in pitch and don't come back down. It wasn't like that before, what should I do to fix this?
The action and keeping in tune are not particularly related. You could adjust the springs that balance the string tension, but it seems like the problem is more to do with the strings binding at the nut end of things.
Lubrication in the form of graphite in the nut slots is always a good starter. It's possible that the strings are binding due to nut slots being too tight (narrow), but with .010"s it's doubtful.
When you press down or pull up on the bar does it stay in the position you moved it to or does it snap back to the home position? If it snaps back then Tim’s advice is on the money. If it stays where you move the bar to then it seems like there is something wrong with the tremolo itself, like it is catching on something or whatever it hinges on is damaged.