I bought my dive computer after I have everything else.
Maybe you can use these as consideration.
1. Wetsuit : I guess wetsuit is the first thing you need to buy for hygiene reason. Remember, divers pee in their wetsuit.
And consider about skin bacteria etc with rental wetsuit.
Mask and snorkel come close as they don’t cost much but good to have your own.
2. BCD, for new diver, buoyancy and equipment stress are big problems. Rental equipments are sometimes in bad condition. I hate my first ten dives with leaking, oversize and half-broken rental BCD. It almost robbed me out of this hobby.
3. Regulator. Rental regulators are almost always in terrible condition. Leaking, spoilt gauge etc.
4. Fins. It’s good to have fins on your own, choose the suitable size and powerful ones to ease your dives.
5. Other accessories, light for night dives by the time you are ready for AOW, whistle, gloves, booties, pouch, smb, slate, weight belt.
It’s better to have safety device like smb and whistle early.
Dive computer is an important factor too. I used to think that I will survive without dive comp. I have my depth gauge and analog wrist watch to keep my safety stop time, when I logged my dives, I used my buddy’s informations.
But it’s not accurate and with dive computer, it’s safer because it helps you to know how long you could be in whatever depth, safety stop, accurate dive profile etc. But it’s not the first priority.
Imho, find out whether you really ready to commit to this sport.
If you are, my advice is to invest in good equipment rather than upgrade later would cost you more. But that’s personal interest and considerations.