short, fat, first time cyclist I'm short 5'2" (1.57 m) and 247 lb (112 kg). I've been working to lose weight and have come down 5 st (30 kg) so far. Now I really want to ramp up my exercise and was looking at cycling as an option.

I've tried looking on websites for advice but can't see something simple I can understand. 

I need some advice on how to select a suitable model that would be a good first time purchase. I'm clearly not going for speed and adventure just yet, so something basic that will fit my size and weight.
You've got two things going against you:


Height: I'd suggest avoiding 700c wheels and going for 650b (pricer) or 26" wheels. For short riders (I'd say shorter than 5'5"), these provide better fit. Surly, for example, only makes their small bikes with 26" wheels. 
Weight: You're going to have to check each manufacturer's weight limit. Generally, hybrids and mountain bikes will be rated up to 300 pounds. That being said, expect more risk broken spokes and rim damage if you hit a road hazard. Cheap wheels are going to be your primary adversary. You also want big tires so you don't have to run them at insane pressure (which can increase chance of wheel damage) to avoid pinching the tubes (unfortunately, big tires increase the standover height, which can be a problem for short riders). On the road (and light off roading), you don't need suspension (and for high weight riders, you're going to load the suspension unnecessarily, especially on cheaper stuff). 


My recommendations are:


Find a bike shop in your area to buy the bike. With high weight and low height, online bike buying isn't going to be very good for sizing or comfort and you'll likely want to swap out some parts on a stock bike anyway (saddle, tires come to mind). And most people won't ride a bike thats uncomfortable.
You may want to buy upgraded wheels (or more likely, upgrade wheels when the ones that come with the bike may start to fail). In general, don't cheap out. Reading Peter White's page on wheelbuilding is a good place to start.  
Look for bikes with no suspension that can take big tires. 700x32 minimum (preferably 35 or 37) if you choose to go 700c wheel sized (though I recommend 26"). 26x2.0" for 26" is ideal. Hybrids, mountain bikes (front suspension is OK but no suspension is better; swap on slick tires) or touring bikes are all good candidates. Cruisers are also not a bad choice, since a lot of them have step-through ish frames which can be easier to get in and out of. Step through and mixte frames in general are good idea for shorter riders (mixtes being stronger than normal step through frames). 
Don't buy a bike just cause it looks cool. 
Men's bikes work fine for (most) women. Most of what makes a women's bike is maybe a slightly lower top tube, a wider saddle and a ridiculous paint job. That being said, some women's bikes do have better specs for women (Trek's Shift line has 26" wheels, and their mountain bikes aren't all 29ers, though this seems to be happening on the mens side too). 


You may want to look at bike forums with sections like "athenas or clydesdales" -- these are typically heavier riders (clydesdales are generally men over 220 lbs, athenas are women over 165 lbs). Example is this one. If you're upgrading components, they may have specific recommendations which may be useful.