If the kick is bleeding into other mics...then EQ out the low end on the other mics. Especially vocal mics. Most boards, even cheap ones, have a button for high pass filter (HPF), which rolls off the low end, or just use the low eq knob. The lowest of male voices only gets down to around 80hz, and most pop vocals are way higher then that. so anything below that just cut right out, and you'll eliminate the kick drum bleed.
If a lot of bleed is the problem then a gate/expander will help you more then a compressor. In very simple terms, a gate on the vocal mic will mute it when the singer is not singing.
The 'trigger' to open (unmute) the mic is a specific threshold of volume. Ideally, a vocalist singing directly into the mic is a louder source then just the bleeding kick drum/other stage noise. Thus, when they are not singing it closes.
That's the basic gist, but Gates/expanders have some more controls to make things more smoother or natural sounding. So you're not 'muting' the vocalist in mid phrase, or causing a stutter effect. Just like compression, it can easily be overdone and make things sound worse.
Some fancy gates have a 'key filter' which can be set to a specific frequency. Then, the gate will open only when that frequency is loud enough. A good example is a gate on a kick drum mic. A normal gate may be unintentionally triggered when the drummer hits the snare. But if you set the key filter to 40hz, the snare drum doesn't have a lot of volume in that range, so it won't trip the gate. The kick drum likes that range however, so it will open the gate. For vocals I like to use 2k as my key filter, which keeps the cymbals from triggering it.