First of all - if you are new to diving, when you do the course, the rental of dive gear is included - you don't need to buy your own to start with. And even if you are certified, you can rent the equipment as well.
What I did though when I first took up diving was buy my own mask, as I found the rental mask were not really meant for Western face with high nose bridge.
Most of the dive shops are selling dive gear.
There used to be a good concentration of diveshops and gear shops in the Concourse, but this building is no more, so they have all relocated to different locations now.
Probably the largest dive (gear) shops are Waikiki and Orpheus "theSCUBAshop". If you are looking for particular brands, like RalfTech, they have their shop, which opens only by prior appointment, just give them a call - they are near Boat Quay. Google search the names for the latest locations.
For dive operators, I had good experience with Planet Scuba Divers. But others like Deep Blue as suggested are good too.
You can also check out the forum
http://www.sgscuba.com/forum to read stories about local and regional diving, or ask your questions.
If you go on a dive trip to Tioman or Aur with a local operator, you will typically travel on Friday night, do 1 or 2 dives in the morning, have lunch, do 2 or 1 afternoon dives, 1 night dive at about 6pm, have dinner, some R&R at the resort, sleep, Sunday morning 2 morning dives, lunch and back to Singapore. So basically there is no time really to do any further sightseeing.
What you also could do if you want to go for a longer trip is arrange your own trip up to Tioman. Go to the main village of Kg. Tekek by ferry from Mersing. You will find resorts and dive operators there. I have good experience with Tioman East Divers and since recently they have their own resort as well.
Don't expect too much of the village though, it's really a "kampung" - a few restaurants etc. But you could probably do a non-diving day and rent bikes or transport to see some of the rest of the island.
If you going to Tioman with one of the Singapore dive operators, you'll be based in the southern part, which is sort of "stuck" in the resort just like when you go to Aur. Not a bad thing, after the diving you really just want have your dinner and relax...
Hope this helps!