First thing: Wall up the breach. If you breached it from above with an up-down stair, just build an up-stair on that tile (this builds a floor so flying monsters won't be able to come into your stairway.
Second thing: Dig a stair or hallway to a level where you can access the cavern floor horizontally, and ideally have some room to fit in an entrance hallway, drawbridge, cage traps. Dig the entrance hallway (except for the last tile), hook up the bridge, load the traps, etc.
Third thing: Set standing orders so that the dwarves do NOT auto-collect webs. Unless you want them all charging blindly into the cavern as soon as it's opened.
Fourth thing: Mine out the last tile. Be ready to close the drawbridge if necessary.
Fifth thing: Up to you, now. Often I'll have the militia explore the caverns by issuing station orders where I want them to go. Meanwhile my peasants will be chopping trees and gathering plants near the entrance. Keep an eye on the (u)nit lists to see if any wild animals enter the map. There aren't a lot of real dangerous ones in cavern 1, but watch out for giant cave spiders and cave crocodiles. Stuff like crundles are good for militia practice and meat, so don't be afraid of letting your soldiers at them. If you see an encampment of sentient animal-men, though, I'd retreat and come back with a good-sized squad and archer support. They aren't too tough but they do have archers and killing them isn't worth taking casualties.