On water bottles, if you buy Ice Mountain water in the 24 oz sports bottle, it fits nicely into almost all bottle cages, and a 24 pack at Sam's Club costs about $5. You can refill if desired, or carry 2, and throw out 1 after riding halfway to lighten the load for the ride home.
It's good to be able to repair a flat tire, and maybe tighten a few things, but otherwise? The likelihood something else will break while riding around town, to the point of rendering the bike rideable is pretty low, and do you know how to repair a broken chain anyway?
You can buy a kit that has 2 tire levels, a few glueless patches, and a couple of itsey, bitsey C02 cartridges that will inflate a few tires so you don't have to carry a pump. This will fix most flats, the whole thing fits into a tiny little case, and you can buy one at Sports Authority for under $20.
The only other tool you really need for emergency repairs is a little "multitool"--such as the Topeak described in a previous post. A small one will at least tight up a loose seat (so you don't have to ride home standing on your pedals!), loose handlebars, etc. I got mine also at Sports Authority for under $10.
Then, you should also get a bag to carry these items, either one under the seat, handlebars, or a rack that attaches either to the seat post or to the back frame.
To me, everything else is nice, but not essential.
Fenders? Maybe you might get a little dirty, but then again, maybe not, and if you have a road bike, fenders won't fit!
Panniers? Gee, I've ridden on 2 week trips and never had the need. Too much weight. Are you planning on carrying a tent and sleeping bags? For overnight trips, credit cards and hotels!
Pump? CO2 cartridges are much smaller, cheaper, work just fine, and have you ever tried to pump 100 pounds into a tire by hand?
Map? Plan out your route and print out 1 page using your computer and "Mapquest".
Lights? Only if you're traveling at night.
Cycling computer? Nice, but not necessary. To me, a compass is more useful.
Biking clothes? I just don't like them and wear my gym shorts.
Of course, to each his own.