what will happen to coins on the other fork? I've received at least one notification about a company that will/will not support the hard fork. does that mean if their side of the fork falls out of favor, I will lose any coins kept with them?
if so how should I move my coins so that they won't be lost?
Move all your coins to a wallet where the private keys are under your full control.
You should not lose the legacy BTC that you have with any service now, but you may lose the new coins if they dont support the new fork.
Once you have the coins in your private wallet, you can get the software of all the new forks aopen your original "legacy" wallet with the new software and you will have the new coins.