For outdoor single-pitch climbing, a pretty bare minimum is:
- shoes, harness, belay device, locking carabiner, helmet, nut tool
This assumes that you're climbing with someone who owns a rope. Harnesses are pretty generic. Shoes are theoretically not necessary, since people climbed some pretty hard routes in the 1930s in hobnailed boots, but realistically they're a huge advantage. The nut tool is needed for trad climbing, so you can clean the gear placed by the leader.
For outdoor multipitch climbing, add:
- Prusiks and slings, headlamp, water bottles that can be clipped to a carabiner
This assumes that you're going to be following someone who knows what they're doing, and that that person owns all the other necessary gear. For the water bottles, I just use half-liter plastic bottles with a loop of cord duct-taped on. The headlamp is in case something goes wrong and you have to finish the climb or descent in the dark.
The Prusiks and slings are for use in case (1) you need to ascend the rope because you have to follow on something that you can't get up, or (2) you're rappelling and need to ascend the rope for some reason (e.g., the rope doesn't reach far enough). You need to practice using these in advance.
All of these are bare minimums if you're climbing with someone much more experienced, who knows how to deal with things that go wrong. From the fact that you're asking these questions on the internet, I assume that you haven't yet made contact with a climbing partner. That could be OK if you're just asking in order to gauge whether you can afford the gear, or if you want to buy minimal gear before approaching anyone to climb with.