You've got two different things to test. One is the upgrade, the other is how your apps perform. It's been a while since we upgraded from 6.5, but the wizard did the work - pretty easy. I'd suggest restoring a backup onto a different machine, running the upgrade to see how it goes. You want to be very familiar with the process, amount of time required, etc, before you do the real upgrade.
Once you've upgraded the text server to 7.0 (or 2000 if you can manage it), then test your apps. If they are internal I recommend you get your develop team involved. Either way someone needs to come up with a list of core tasks you can test to make sure everything works. Only issue we found was a minor bit of code that did concatenation (behavior changed in 7 so that nulls propagate). Setting the dbcompatibility back to 6.5 is an easy fix until you have time to fix the problem. For legacy apps that you CANT fix you can just leave in 6.5 mode, take advantage of 7.0 features for other dbs.
It's a lot of work, but well worth it!