I was going to start boxing, and also playing basketball or soccer. But then I found out that it's bad to bind while doing those things. Is there any way to bind while doing those kind of things without hurting yourself? like any good binders for it, or alternatives to binders.

Sports bra? That is, after all, what they are made for.

When I used to bind the ace bandage method I could not do anything active without feeling the pain or loosing the ability to properly breathe.  The only binder I've had is one I've purchased from T-kingdom and I've really had no problems working out for hours at a time in it.  It's admittedly a pretty expensive investment, though it's been totally worth it for me.  Sports bras don't cut it for me.  As a side note, getting hit in the chest area when wearing a binder is much less uncomfortable than when wearing bras for some reason.