it's a nice (and cheap) option to look for a used hybrid as your first bike to see if you like cycling... i'd recommend getting a 2-3 year old trek fx series - this is their hybrid line. i've owned 3 bikes from the series and they're all great. you can find deals on these bikes for around $300ish.]

i'm 6'1" as well and can ride either their 20" frames or their 22.5" frames. personally, i feel better on the 22.5" so that's what i currently have. before you buy, get the size and take it for a test ride - you can also do this at the lbs. that way, you can see which bike/size you are more comfortable on.