Congrats! You sound like me when I got certified.
What's next? I would try and get some diving in right away. That way you don't forget about the skills you just learned and you can get comfortable.
Owning your own equipment is a good idea, but don't rush out and buy everything right away. If you are ready to spend some money here are my top suggestions: 1. Wetsuit... because wearing a rented wetsuit is kind of gross. 2. Computer... It's easy to travel with and you'll always know how it works. If you rent one, who knows what model you'll get.
This is assuming that you've already purchased your mask, snorkel and fins.
I know that I might get my dive shop owner card taken away by saying this, but try out a couple of BC styles before you commit to one. Most instructors put you in a jacket style BCD, but give a back mount BCD a try. Some people, including myself, prefer to have the bladder on their backs instead of wrapped around them. The good news is that many manufacturers, such as Aqua Lung, have a satisfaction guarantee program. If you buy it and don't like it, you can exchange it for something else. Some dive shops, such as Force-E, have similar No-Worry buying programs. Ask your LDS about their policies.
As far as regulators are concerned...Owing your own is ideal, but if you need to budget your new dive addiction, then renting at first is ok. Most dive shops do a good job at maintaining their rental lockers, but inspect it before you walk out of the store. Hook it up to a tank and breathe off of it to make sure there are no problems.
Best fishes,