is it a good idea to save my lawn bags full of weeds and branches in the garage, let them dry, burn them, and then use the ashes as fertilizer for all of my plants?

Only if you need to bring the ph more basic. Test your soil first. If it is really acidic, wood ash can bring that more neutral. You should always test first though. My garden area is right under pine trees and gets needles dropped every year on it. I had thought it would be very acidic. I had planned on mixing in wood ash. I tested first though, mostly on a whim, and found out it was a 7 ph. Not acidic at all. Putting ash on that could've been bad.

how do you test the ph?

You get a soil test done or buy a soil pH meter. The former is th best option. You could try the nearest OSU Extension office, or here's another option in Cincinnati:

http://www.hcswcd.org/soil-fer...

Personally.... I would pile the branches out of site... Or in the veggie garden, and then pile the weeds and other clippings on top, and allow it to compost.
Far more product afterward. Very little left after burning, hardly worth doing.