To zone with Peacock you generally want your opponent around the max distance of George At The Air Show. If they're closer than that, run away using air backdashes, j.HK, j.HP, teleports, and assist calls until you find yourself at your ideal spacing. If you're using Squigly, a good Squigly assist might be Drag 'n Bite, since it puts your opponent in a lot of blockstun and pushes them back, giving Peacock time to escape.
When zoning, always throw out 1 or 2 Georges before you do anything else. Lots of Peacock players like to start zoning by using BANG BANG BANG or charging an item drop. The problem with BANG BANG BANG is that it does nothing to cover the vertical space; your opponent can easily jump over it and hit you for a full combo. Item drop is a good move but charging it does nothing to cover space immediately, giving your opponent a chance to get closer. If George's Day Out isn't on the screen, getting it on the screen should be your first priority, it controls the horizontal space for a long time and recovers quickly. George At The Air Show is good because it stops your opponent from jumping in, although it will whiff if your opponent is either too far or too close (this is why it's so important to get the right spacing with Peacock). Boxcar George is the best for setting up full screen teleport mixups but I don't really use it for zoning unless the opponent is too far away to hit with George At The Air Show.
Once you got your Georges out you can continue to zone with s.HP, item drops, BANG BANG BANG, or assist calls. You generally want to set up a rotation with your projectiles where the cooldown on one ends just as the cooldown on another begins; a simple one is George's Day Out canceled into George At The Air Show, s.HP, H Item Drop, repeat. If your opponent finds a gap in your projectiles and gets closer, run away until you're back at a good spacing.
As far as rushing down with Peacock, I have no idea. Best advice, watch this guy's channel and copy everything he's doing:
http://www.youtube.com/channel/UCApJgsoFadTZjesGtS-fgVA