I'm going to ignore certain people at the gym
What makes you want to try to ignore them?
They keep egging me on to tell them the reason.
Ignoring people may help you stay positive without their negativity affecting you. As long as you're respectful to the other people there, you can ignore them and go about your business by yourself.
I'm not going to be friendly to them. I just have better things to do.
I think that you should just be respectful towards them. You do not have to talk to them, but you definitely should not be rude to them.