If you search the web for "repair manual" plus the model of your drier, you may be able to find a copy of that, or a site which will sell you a PDF if it for a few bucks. That will give instructions on how to disassemble and reassemble the drier. To access the timer, that's usually a matter of pulling the knobs off and loosening some screws on the back so you can remove the faceplate -- but it's a lot more comfortable when you're sure which screws to loosen.
Actually diagnosing it and fixing it is probably best done by unplugging and dismounting the timer from the drier and working with it on a bench, if you're comfortable dealing with house current and figuring out which leads power the timer unit and which ones get switched on and off by it. There are websites which discuss that too. You're lucky that this appears to still be a mechanical timer; if it had electronic controls they generally are pulled and replaced as a unit rather than trying to diagnose them.
If you're lucky, it might just be that the timing motor has failed. If so, that's usually a standardized part and may be attached with screws and thus be replaceable. The hard part would be determining what speed and direction and output gear is needed; the manual may have that info. I'd start by seeing whether the motor turns at all; if not I'd dismount it, apply power to it, and see if it now turns; that will tell you whether the problem is a failure in the motor or the gear train.
The manual may, or may not, tell you enough to test this properly. Sometimes there are interactions where other connections have to be powered or not powered, eg if the drier has a moisture-sensing cycle. (There's usually a straight timed cycle; that will be simplest to check out.)
Basically: If it's already dead, there's no reason not to play Dr. Frankenstein and try to bring it back to life... but ONLY if you're sure you can do so safely. If in doubt, finding the right replacement timer and swapping it in, as @Mazura suggested, is comparatively straightforward and might still save you the labor costs.