Really wanting to do this on a budget looking for some info

If you have a decent pair of shoes and a bike, you're 99% of the way to duathlon-ready, gearwise.

If you have any specific questions, I'd suggest posting them in the Triathlon Talk forum. Welcome to BT!

Welcome, and congratulations on the weight loss. 

I'm a little confused...do you have a bike (even an old one) that works?  A 15-mile ride isn't too far (an hour, more or less) with even a little training. 

If you have an old bike, check the quality of the tires and tubes (as they get old, the rubber deteriaorates).  New tires/tubes aren't too expensive depending on the size of the tires.  Check the brake pads too, because that rubber also gets old and you really don't want your brakes "going away" on you.  A little WD-40 for the lightweight moving parts (like the derailuers) and something for the chain.  If you're ambitious and mechanically inclined, it isn't too hard to re-grease axle bearings, bottom bracket, and head bearings, JUST TAKE YOUR TIME TO GET IT BACK TOGETHER RIGHT.  There are some good on-line resources (sheldonbrown.com for one, park tool for another).  Work slowly, don't "force" anything, and make small changes that you can undo if you don't like the results (so...if you adjust a deraileur, turn the screw 1/4 or 1/2 turn and check the results.  If you're not happy with them, turn it back the 1/4 or 1/2 turn and adjust something else.  Even if you really muck up the adjustments, you can usually take it to the local bike shop and have them give it a "tune up" before you really damage something. 

You'll need an approved helmet.  Besides the bicycle, helmet, and running gear, you really don't *need* anything, but having enough stuff to change a flat tire would be good.  A way to carry water (either a water bottle or a camelback) on the bike is also good, and I'd probably wear sunglasses to keep dust and bugs out of my eyes.  You can skip the bike shorts, clipless pedals, shoes, etc. for a while if you're trying to keep costs down.  I did my first tri on a mountain bike, wearing baggy swim trunks, an old cotton t-shirt, and running shoes.  None of that kept me from having a blast and I've stuck with the sport for 4 years now.