Welcome, Cedar! It sounds like you've got a good handle on how to approach a mostly-unknown stash - grab something to start with and get going! I love that you're finishing up some of your grandmother's unfinished projects - I think she'd be so pleased. That being said, it also sounds like you're pretty new to the quilting game and so you may not have a really clear idea of what you will most enjoy doing or how your interests and abilities will evolve.
If you need to get rid of some of the stash quickly, you might want to do a quick scan through and find fabrics that really don't appeal to you at all and start putting those in a tote or a bag for donation. If you look up local quilt guilds or organizations like Project Linus, they're usually able and happy to use fabric you can't. If you have smaller pieces of batting, some guilds make placemats for Meals on Wheels and they might be glad to take those, too.
If the stash is really random (as mine is), you can also look for fabrics that seem to go together well and could make a future project and start putting those together in stacks or bundles or boxes. You might notice some fabrics just don't seem to want to "play" with anything else there, and that can be another good way of weeding out some of the bulk.
I'm sure there are lots of other ideas out there, but hopefully that will help you get started. Good luck - I hope you are able to come back and report on your progress soon!