Regarding type of bike - the clue's in the name. "road" bike = riding on the road, "mountain" bike - well you can work out the rest.
Pish taking aside, the above is basically true, but there's a wide range within each broad category.
Starting with mountain bikes - these have suspension and fat nobbly tyres and are quite heavy unless very expensive. Nobbly tyres and suspension are a really bad thing on roads and unnecessary on moderate tracks. if you're mainly road cycling you do not want one of these !
A "road bike" proper, what we used to call "racers" when I was a kid, have dropped handlebars, no mudguards, light weight, and rather high gears. You can get quite a lot for your money in this category, but I personally like low gears, mudguards, and a pannier rack.
Which brings us to the Tourer, or Audax type bike, still with drop bars - slightly heavier than the above, but will have the lower gears, mudguards and panner rack - an excellent all round choice. Snag is they tend to start at a higher price - not that the concept ie inherently more expensive, but they tend to be made and priced in a higher-quality bracket
There are also so-called hybrids which are a bit like mountain bikes in that they have low gears (a very good thing), flat bars (a bad thing in my view, but to be fair flat bars do suit some), and mudguards (another very good thing), but don't have the weight and suspension of true mountain bikes - which you absolutely don't want for road or moderate tracks.
Some of these are pretty good value, but I personally don't like flat bars - but that's my preference, so is a personal thing.
Don't get a hybrid with suspension if road riding - more work, more weight and a bad thing.
With a tourer or hybrid, having slightly wider tyres and so on, you can get away with the odd track as well, without all the excess weight of a mountain bike proper.
There's also the cyclo-cross bike which is a heavier built "road" style bike that can take a lot of knocking about off road. Not a bad choice even used mainly on road, and the gears will be a bit lower than on a road bike proper.
Regarding your weight, I shouldn't worry one bit - at least not cycling wise, as it's not that bad, unless you go for ridiculously light flimsy wheels and ride up kerbs. You'll get fit, and may or may not lose weight as a result, but fit is the main thing.
Oh, and don't buy a padded or gel saddle
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