If you found a cave system you can fix it up into a lair, a good one can make a base for the rest of your game. If it ends, then cap off your entrance. If it does not end, then build a little shelter around you [usually by capping off both open ends]. Don't worry too much about the natural walls of the cave, zombies can still be heard through the walls pretty loudly, which may be a hint where not to dig.
To block the cave off, you need a bit of thought beforehand. If you use a door, you need a back door as well (and maybe some windows), in case a creeper is hanging out by your usual exit. Possibly a better way to block off a cave entrance is to run a fence with a gate across the entrance. Clear away any stray blocks by which monsters could jump the fence, and extend the entrance ceiling to foil "drop-ins". You also want at least 5 or 6 meters clearance inside the fence.
That way, if a creeper comes up to the fence, you can pick it off with a bow without setting it off. (Other mobs can be treated much the same way, but creepers are the most serious hazard.) What you do not want, is to come up a ladder to your exit, and find a creeper 2 blocks away on the other side of the fence.
You can dig into the cave's wall while waiting in either of these cave situations, but keep building blocks handy to patch up any openings you might make into another cave while digging, which might have a hostile mob in it.
If you place your pickaxe in the quickslot bar (used to navigate quickly between items you are carrying in your inventory) and e.g. dirt is placed right next to it, you can quickly switch between your tool and that block type in order to close the hole you just made. This can be extremely handy when you stumble upon an opening which contains mobs.