I have an old ten speed, a Trek 1500 road bike and an Electra Townie... my husband is an avid cyclist. I have never had a flat tire and biked when I was even heavier than I am now. The Trek 1500 was the bike I used on a 100 mile bike ride, I weighed in the 230's I think. No flats for the entire time-- BUT- I may just be really lucky.
It does have the skinny butt seat, I am embarrassed to sit on it now, to be honest. I think from the back it probably looks like I am sitting on the seat post.
Now I ride my Townie as I have a bike trailer on the back for my daughter. It has thick tires and, if you are not a great bike handler (I am not,) it is great because it is virtually impossible to fall off of. It has 'flat foot technology' meaning you can always put your feet on the ground. I have a 21 speed so I can use the 'granny gears' on hills. It is the bike I got when I was pregnant so I could continue riding and feel safer, I am clipped into the Trek pedals. You may want to look at hybrids as well. You can go farther with a hybrid than a townie because the tires aren't quite so thick.
Team estrogen is a GREAT SITE! (robin41 recommended it as well.) It has a lot of great information and was the only place I could find plus-sized cycling gear. Doesn't mean I looked good but it was nice to have extra padding.
(I just wear jogging pants on the townie.)
I hope that is at least a little helpful. I LOVE bike riding, we have already been out this morning for 30 minutes. That is as long as my DD will last so I take it. Good luck!