What I have repeatedly found with GA is that I can solve some problem I have using some other brute-forceish tools with a few pages of tricky error-prone scratch-work that balloons out to a complicated mess before simplifying back down at the end, and then afterward think about it a bit and come up with 2–6 lines of simple GA identities showing the same thing in a much higher-level coordinate-free way, and with most of the steps geometrically interpretable, rather than just opaque calculation. But coming up with the simple version at the beginning is hard.
The tricky part about it is that there are a lot of useful identities that can be written down, and properly learning a decent number of them and figuring out which ones to apply in which situation takes probably years practice, ideally with some guidance/support from someone who knows more than you. (I do not feel like I have mastered the subject.) The same thing happens using whatever other formalism, with the difference that many identities that are pretty short to write down in GA are much more complicated to write down, so people don’t even try to use them.
I’m not sure if there’s really a good beginner source, but I haven’t ever really sat down and tried to go comprehensively through the exercises in any books pitched at a relatively elementary level. You could try Alan MacDonald’s book Linear and Geometric Algebra which is designed as an intro undergraduate textbook. If you want to also learn some mechanics, you could try Hestenes’s book New Foundations for Classical Mechanics.
