There is a huge amount of information on the net with regards to amateur robotics.
First, you have to set an upper budget limit, as that will define the sort of technology you can use. There are too many sites and topics and subcultures to list here.
Briefly, here are three rough categories for the way a robot can be controlled:
1. Microprocessor
2. Microcontroller
3. Non-CPU feedback mechanisms
Microprocessor-controlled robots tend to be the most expensive, though things are changing. Some devices can be very powerful, in effect having a PC control the robot in a relatively small package.
http://gumstix.com/
Microcontrollers used to be cheaper, but their price doesn't seem to be dropping as quickly as microprocessors. They are also not as powerful as microprocessors.
http://www.hobbyengineering.com/SectionRK.html
http://www.hackaday.com/entry/1234000487032939/
Simple feedback mechanism robots are very simple, but are difficult to design to do sophisticated tasks.
http://en.wikipedia.org/wiki/BEAM_robotics
Stair climbing robot links by searching google with
stairs robot hobby
http://www.robots-dreams.com/2006/03/up ... tairc.html
http://www.botmag.com/articles/vex_1.shtml
http://www.diynetwork.com/diy/shows_dro ... 90,00.html
One of many sources of materials for wheeled robots.
http://www.acroname.com/index.html
I strongly suggest you spend a while searching Google and refining your keywords. You will probably need to spend between US$100 and US$200 to create a device that can reliably climb stairs. Keep it simple, don't get over ambitious, and do not expect to create a powerful, versatile android!
If you are clever, think about things, and above all try several different strategies, I expect that you will be successful in building a relatively inexpensive device capable of climbing stairs.
I can't emphasize this enough:
Keep it simple!
Set only one goal: climbing stairs
try to use cheap materials instead of expensive ones
research research research
don't be afraid to recycle ideas you find on the net
use your imagination
Keep it simple!