First of all practice. Dedicate an hour to taking atire off and putting it back on. That way when it has to be done on the road when you're in a hurry it will be easier.

I will assume you have clincher rims. Get an extra tube to carry with you. Two tire levers, One patch kit and a pump. The tube is self explanitory. Just pick up a $5 tube to always have with you.

Tire levers are what you use to take the tire off the rim. you probably only need on but it doesnt hurt anything to have an extra.
Get a patch kit that has patches that will work on both the tube and the tire. That way if you cut your tire you can patch that also.
For the pump you have have 2 choices. Either a frame pump that attaches to your top tube or screws into one of water bottle holders (the water bottle cage is still useable with the pump). These pumps you have to manually inflate the tires. it takes longer than the next option but you never run out of air.[URL=http://www.trisports.com/inroairfrpu.html]Here is an example[/URL]
The other option for pumps is one that uses CO2. The advantage to this is you can inflate the tires to higher psi (frame pumps only good to about 90).They also inflate your tires quicker than by hand. The downside is once you run out of CO2 cartridges you're done. If you do use CO2, dont buy them from the LBS. Go to your local Kamart or Walmart and hit the gun section. Its much cheaper there. [URL=http://www.trisports.com/ulihpro.html]Here is an example[/URL]

Lastly I would suggest it might be cheapest in the long run to buy something like [URL=http://www.trisports.com/intireinwa.html]this [/URL] or [URL=http://www.nashbar.com/profile.cfm?category=99&subcategory=1232&brand=&s... which has otehr tools you will eventually need while on the road.

Mike, this is a really helpful post...Thanks! I am new to the forums. Distance running is my strength. I'm a newbie cyclist and swimmer. Your posts are great!