I agree with Live4sharks, wetsuit is something I bought first for hygiene reasons! Also it is much more comfortable to have one that is fitted correctly than one you hire in a shop (and I dive in coldish water so wanted a semidry, which isn't rented out at many shops) I bought the rest together, but if had I to buy separate I would have then bought reg, computer, BC, weights and tank in that order.
My reasoning for this is that regs are really important and I found rental ones started breathing harder once I got to 20m. Also again, there are hygiene reasons with sharing regs!
Then computer to go with my regs as I got an air integrated console one. I liked the idea of having a computer I knew well and that I could upload data to my computer, but that is a personal preference to me.
Then BC, I used a few types of different BCs when renting and found them all good enough not to have to worry about them so it is fine to take your time and find out what is really comfortable. Also there are a number of types so you should decide on what kind of diving you are going to do before you get one. Like my BC can't really take doubles and eventually I want to get into wreck diving so will probably need a BP/W set up. I got a backinflate BC for example as I found it much better than a jacket style. It is less expensive if you wait and see where you want to go with your diving than buy two BCs. I know you said you want to start with the BC, but I would really advise against that personally or if you still do, try out lots of types! Others may disagree with that though...
Weights and tanks, last priority. They are easy to hire and don't need a good fit so low priority.
But yea, just my 2c.