I second the specialized shoe recommendation. I got a mtn bike stiff-soled shoe (road shoe with treads on the bottom) for $75 from a local bike shop.
As to priority of add'l items, I would put them as follows:
1. helmet
2. glasses (wind etc. in your eyes can limit your visibility and thats a bad thing.)
3. Flat repair provisions (seatbag, tubes, co2 or pump, patchkit.) BTW, if you get a pump, make sure it can handle the high pressure required to inflate road tires. (I learned the hard way).
4. (tie) padded shorts (baggy or tight)..Either way, you will have a sore bum for a bit. The padding helps keep the pain down to a tolerable level until you acclimate.
4. (tie) cycling gloves (esp. at first. They help you maintain grip when you sweat)
5. Shoes (stiffer sole improves efficiency)
6. Pedals (I like spd's even for road use. To start I recommend one of the mtnbike platforms with the spd cleat inside. That way you can ride the bike with or without the shoes.
7. Computer. You don't need a computer to cycle. It certainly helps you keep track of miles and speeds etc, but for the most part, it doesn't matter a whole lot. When you do get one, I suggest one with cadence. I particularly like the Astrale-8. Cheap, has cadence, is wired so no interferance with HID lights or HRMs.
8. Jersey/non-cotten activewear shirt. You can start off with t-shirts just fine (I rode my first 800 miles before getting a jersey), but make sure the weather is warm since you will sweat and then the shirt will not help you retain heat. Otherwise, wear a windbreaker or a wind resistant pullover to help keep you warm.
I also agree with others that you could easily spend over your budget, and that you will have to do some smart shopping to get in on budget. If possible, I'd up your budget, or I would buy in phases: 1-3 first, then the two 4s, then 5&6, then 7&8 etc to spread out the costs but still keep you out there in the meantime.
Welcome to cycling. If you haven't already, try to find a local bike club and go on the rides. Its a great way to meet people and learn about the sport.