I concur with the wetsuit/bc/reg/computer/tank order of purchase crowd.
UNLESS you are one of those people who get the heebie-jeebie's out of putting a reg in your mouth that has been, at one point in time, in someone else's mouth, even though it has since been washed thoroughly (wash it yourself to be sure, if it concerns you that much). If you're that paranoid, then you won't be happy until you own your own regs. One thought: I've seen people buy their own mouthpieces and detach the used one. Might help if you're one of the overly-squeemish.
And also, in deference to JodiBB's point, however, I can agree that if your LDS is not renting you quality gear (it is your responsibility to check your gear thoroughly before allowing your life to depend on it!), and their rental equipment is known to have problems, then again, it may make sense to switch the BC/reg purchase order. If that was the case, however, realize that you can get into just as much trouble with a malfunctioning BC as with a malfunctioning regulator.
All that said, as long as you can live with rental regs for a time (we're not talking about forever, just next in line on the purchase plan!), nothing will help your diving more than working on buoyancy control, and for a new diver especially, nothing helps that more than having consistent gear so you have the chance to do repetitive dives with the exact same buoyancy characteristics, allowing you to dial in your weighting and get comfortable operating your BC at depth, clipping things to D rings, finding things in pockets, etc.