The main problem I have with geometric algebra is that it's an alternative framework to express exactly the same things. Some things are expressed or derived easier in G.A. e.g. the Kerr solution of General Relativity, electron Zitterbewegung, etc. The problems with G.A. as I see it... 1) It's a specialized tool that does not generalize well to simple everyday physics problems. Don't get me wrong, I like G.A. and learned it from Hestenes' papers almost 20 years ago. I just don't see the point in trotting out G.A. to solve certain problems or illustrate some theory any more than I would abandon Gibbs vector calculus and insist on doing all calculations in 4-vector notation using field tensors and differential forms. I wouldn't even want to use it to teach differential geometry. Once someone knows these subjects I'd say go for it and learn G.A. It's actually very easy once you know the more conventional stuff.

2.) I don't know how things are now but in the past G.A. had the feeling of a cult theory. Unlike most cults, this one had substance. The problem learning the material first with G.A. is that you won't be able to understand other papers, books or solved problems of others. You'll be dependent on using G.A. and unable to make sense of Goldstein or Spivak.

3.) I'm all for alternative ways of doing things. If you have time to burn there's a "Bird Tracks" method of doing calculations in Lie algebras which is very different than the traditional Cartan matrix, Dynkin diagrams. I advise you to avoid learning these topics the first time around in a weird mathematical language. After you know you're way around these topics go for it.