look up CD4026 on the net. The data sheet should probably give you enough info.
If the clock needs to be accurate, you will need a crystal, (or if accuracy is not an issue, a 555 timer IC would do) to give the pulses, which will feed into a 4026, which would cascade into another 4026. Each 4026 will drive a 7-segment display. You would connect your push switch to the 'enable' pin on the first 4026. I have a digram of this lying around somewhere, I'll try and dig it out for you.
The above is probably the simplest way to do it, however, it will only count in seconds. If you want minutes as well, it gets a bit more complicated. However, it would be accomplished using (instead of each 4026) a 4510 and a 4511. The 4510 has a binary output, to drive the 4511 (which drives the 7-segment display). Logic gates can be connected to the binary output and reset the whole thing when it gets to 59 seconds (using the first method it would count up to 99 seconds).
Is this any help?
Tim