Mask, snorkel, fins+boots, weight belt. And a few safety items like a KNIFE or EMT SHEARS plus SMB and whistle. Wetsuit if needed. Now you can at least snorkel effectively any time, anywhere and you have the basics required on every dive.
Then get a regulator set. Less icky than rentals and probably more reliable, but some people will argue that point either way. Then BCD or wing, computer, and tank(s).
If you dive locally with any frequency, buying your own full set of gear (even including tanks) sooner instead of renting makes a lot of financial sense. Especially when you consider that it's often difficult to return rentals within a day and the multi-day charges rack up. It really is worth it. Wish I did sooner.
Do a lot of research, look at what your EXPERIENCED peers use, and try to get gear that makes sense for your diving now and in the future. Don't buy into the gimicky "cool new feature" stuff. Go for simple and tough gear. Everyone seems to regret at least one big dive purchase made early on. Personally I got a fancy jacket BCD because that's what the dive shops rent and sell noobies, but I really wish I bought a backplate and wing right off the bat. Split fins are a common regret around here.
A lot of people will recommend a computer immediately, and it's a valid recommendation, but they are pricey and you can absolutely dive safely without one. Learning to use tables will serve you well. I didn't get a computer until I started doing deeper, longer, and more frequent dives after about ten years with my c-card. Computers are useful, but not strictly necessary. Especially for the short shallow dives a new diver is likely to be making.