There are really only three things you need when you start cycling: a bike, a helmet, and bike shorts. No, not just some ordinary shorts, bike shorts with some padding. They will be the difference between a 30 minute ride and a two hour ride.
I'm definitely not an expert on what kind of bike you should be riding. The only advice I will give is to make sure you get fitted for your bike, whatever bike that may be. If you live near an REI I highly recommend going into a store and getting measured for a bike. REI is awesome. They're happy to help, you're not obligated to buy anything, and they're full of useful advice. Getting fitted for a bike basically means you get measured so that when you buy a bike you buy a bike that is the right size.
For example, my husband and I ride the same bike (different year versions, but the same model), and as you can see, he being 6'2" rides a much larger bike than me at 5'5".
Getting fitted will also help with injury prevention because you will be properly sitting on your bike and not overextending yourself.
Get a helmet. I don't care if you don't want one - get one. They are literally lifesavers. Most states have laws that require a rider to wear them if under 18 years old. But be smart about it - if you're riding a bike, you should be wearing a helmet. Researchers from Boston Children's Hospital found that bicyclists who wear helmets have an 88% lower risk of brain injury. You wear a seat belt when you're in a car, wear a helmet when you're on a bike.
I like this version from Giro that has vents (definitely helps when you start to heat up on a ride), and has a removable visor. I like the visor on sunny days - it helps keep the glare out of my eyes, but its nice to have the option to remove it.
There seems to be a billion different versions of bike shorts to choose from. You can get shorts that are built in to another pair of shorts, you can get stand alone shorts, or you can get shorts you wear under your clothes. I started out wearing the shorts under my clothes because I wasn't a fan of walking around in tight, tight pants when I wasn't on my bike. I went through a couple of pairs until I found the ones I like the best. I love Canari Cyclewear Women's Pro Gel Short Padded Cycling Short. My favorite thing about them is that they don't have elastic binding around the legs so I can easily wear them under shorts or capris and you can't tell that I'm wearing them. They're also super comfortable for the few brick workouts (bike then run) I've done.
I'll most likely be investing in a pair of tri shorts for the triathlon. Marci (of The Marci Files) wears these from Zoot and likes them.
REI has a great selection of bike shorts. Usually its best to go and try on a bunch of pairs to find the ones you like best.
*NOTE: this gear is expensive at full price, but I've always been able to find it on sale at REI or Amazon. Watch the sales online and in store and you should be able to come away with some great deals.
Bike gloves are useful for longer rides. Sometimes you can get blisters on your hands from gripping the handlebars for a long time, and bike gloves can have an extra cushion in the palm to alleviate that pressure. They can come fingerless for warmer days, and full fingered for cooler days (definitely helpful to keep your fingers from going numb). If you don't want to buy bike gloves you can also use weight lifting gloves.
Many cyclists wear clip-in cycling shoes when they ride. I don't. My bike has the clip-in option on the pedals, but they also came with cages. I use regular shoes when I ride and use the cages as the assist when I pull up with my foot during a pedal rotation. Since I have had no experience using clip-ins, I defer to this article by REI titlted: Bike Shoes: How To Choose.