How do I replace the hard drive in my circa 2008 iMac?

If you're talking about an aluminum iMac, getting to the HDD involves removing the built in display monitor to be able to get behind it. YouTube seems to have several video walk-throughs that show the process (including this one; the link is to the first part). Additionally, a quick search turned up a site with written, step by step instructions. The process seems too complex to provide here.

If you're comfortable with this level of complexity in hardware repair, then the video or text walk throughs should help you. I suspect, all things considered, if you're not confident enough to perform the upgrade yourself just by looking at videos of others who have done it, then this probably falls into the class of "take your iMac to your local Mac service centre and get them to do it": yes, it will cost you money, but they can also guarantee their workmanship...

Yes, this is hard to do. Go to iFixit.com and look up the repair guide for your particular version of the iMac to see just how difficult.

You can use any 3.5" SATA drive in the iMac. Choose a replacement drive based on your criteria such as capacity, price, manufacturer's reputation, etc.

The drive should report its S.M.A.R.T. status in the Disk Utility application. If this status is good, I wouldn't bother replacing the drive.