Delete the catalog file and the previews folder that goes with it.
If you work with a defaut configuration, the catalog is named "Lightroom 3 Catalog.lrcat" and the previews folder "Lightroom 3 Catalog Previews.lrdata". See here for default locations and names.
Upon restart of LR, you can choose to create a new default catalog.
There would be other ways also, but I think this is the easiest if you really want to start all over.
My catalog has too many missing photos, probably from moving the photos around since trying a demo LR3.
I would like to just erase my present catalog and start all over again. Which would be the best way of doing this?
Before you do that, why not just delete the missing photos (run find missing photos) and then synchronize your main image folder (the parent of all your images). That will find the missing images if you have moved them elsewhere.