I went through a similar situation of discovering cycling, dropping weight (about the same as mike12 above), starting with the bike I had and then growing into the sport. Where I differed though is I started on a mountain bike, but found that I wasn't riding as much as I wanted on the trails (winter, bad weather at other times), so I bought a second set of rims/tires (with slicks) so I could ride on the road too. This worked well for a while, until I found the road rides lacking with the mtb, so I bought a road bike. You'll know when it's right. I was up to 50k on my mtb bike on roads. I think it's more about the lure of seeing road bikes everwhere that got me hooked. Funny thing is years later and I just bought a mountain bike after not owning one for 8 years. So much better at that now that I'm fitter and stronger. And it's just as much fun.
You could keep your eyes open on the used sites for something that fits you and is a good deal...that way you're not dropping too much coin (until you're ready too).