Okay for the first thing you want to know what you are going to be working with. Lots of things can qualify as a computer. Anything from the antiquated Cosmac, to the modern AM3+ motherboard with everything inbetween. Some things are large and easy to work with, and other things are small and will require a little more precision and care. All the same though you will require some standard equipment reguardless of what you are working with, and then a little extra for the precise instruments.
1. A soldering iron with adjustable temperatures and multiple tips.
Adjustable temperature crucial for working with different types components and different types of solder to make proper joints. Also the different tips will allow for you to do many different jobs. This can range from tight spaces and small components, to large parts and pieces which have high thermal mass.
2. Flux
Flux removes the oxidation and allows for a great soldering joint to be formed. This comes in many forms, and I personally use the one that comes in a can. IT MUST NOT BE ACIDIC PLUMBERS FLUX.
3. Cleaning compound such as Acetone/Xylene
This will allow you to clean up any residue/solder before and after you solder for a proper and professional solder that looks/feels nice, and allows for easier touch ups in the future.
4. Solder
You will want to start out with .5mm 60/40 for its nice forgiving melting temperature, proper size for most jobs and its nice price point for a beginner.
5. a fan
This just helps with pesky fumes that occur as you're working.
6. Cheap multimeter
Check to make sure your work is solid and it doesn't have short circuits in it that would fry components.
7.Solder removal tools such as wick and suction
This will allow you to touch up work with too much solder on the component, and fix solder made shorts.
8. An understanding of electricity and circuits
You should know basic circuit design, and properties of electricity and its place in electormagnatism.
9. Curiosity and determination
Soldering isn't hard, but it can have its frustrating moments. You have to want to continue with what you're doing, and strive to learn even more about soldering.
Some great tutorials were made in the seventies to train workers. The videos may be old, but the principles and methods still apply to todays needs. (middle click links with your mouse wheel to open in a new tab)
For the bare-bones basics
http://www.youtube.com/playlist?list=PL926EC0F1F93C1837
For proper repairs
http://www.youtube.com/playlist?list=PL958FF32927823D12
Intro to surface mount
https://www.youtube.com/watch?v=ihoX7x0RBz8&list=PLB3016C7201738971&index=27
Some additional tools for repair and surface mount device work
1. A table stand magnifying glass
Helps you see individual components/leads that come with 04 series smd devices
2. A low rpm and high torque hand rotary took
helps remove any coatings that get applied to the board so you can reach the components.
3. Plumbers putty
helps with SMD's to stay in place as you work.
With this you have a starting place to begin working. I hope you enjoy the world of soldering.