Specifically, go check out the Suzuki GS500F, the kawasaki Ninja 250 or 500. If you end up with a cruiser, check the kawasaki vulcan 500 (same engine as the ninja 500), the yamaha V-star 650, Honda Shadow VLX 600, Shadow Spirit 750.
For the sporty side, I'd personally look at the GS500F. You can embarrass a lot of bigger bikes on that thing in the twisties if you get good enough.

Actually, you might like the Shadow Spirit.. It has more of a "drag bike" type look with the drag bars, it makes about 45hp, so it should be good to start with, and there are a lot of performance upgrades to make it faster when you feel comfortable doing so. I was looking at them for a while, but I ended up getting a cheaper older bike


As for your question, I weigh around 220lbs, am 6'0, and I started on a 250cc bike. It hauled me around just fine, but I did want to trade up after my first riding season. It was great to get the confidence and experience though, but I think the same can be done on the bikes I mentioned, the horsepower output isnt too much different.