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You might start with teasers like Hestenes’s papers [“Reforming the Mathematical Language of Physics”][oersted], [“Grassmann’s Vision”][gvision], etc.
For a summary from a mathy perspective, try Chisolm’s [book-like thingy on arxiv][chisolm]
If you want a whole book of concrete problems to solve at an advanced undergraduate physics student level, try Hestenes’s [New Foundations for Classical Mechanics][nfcm]. This is the best source I’ve seen anywhere about understanding complex rotations in mechanics problems.
If you want to solve some plane geometry problems, this thing looks fairly accessible [Treatise of plane geometry through the geometric algebra][planegeo] (I haven’t looked too closely).
Two websites are [geocalc.clas.asu.edu][asu] and [geometry.mrao.cam.ac.uk][cambridge]. Also see the [link page at The Net Advance of Physics][netadvance], and this [mirror of Lounesto’s site][lounesto] (he passed away a while back). Lounesto liked to publish [collections of counterexamples][counterexamples].
There are a number of journals, conferences, etc. Try a google search for “Clifford Algebra”.
If you want an introductory undergraduate textbook which tries to teach both geometric algebra and traditional matrix algebra, you could look at [MacDonald][]’s [Linear and Geometric Algebra][laga]; I don’t think Hestenes is really on board with this approach, but there’s not too much else pitched at a similar audience. MacDonald also has a book [Vector and Geometric Calculus][vagc] which I suspect will be substantially easier to work through than Hestenes and Sobczyk’s [Clifford Algebra to Geometric Calculus][cagc]. (I haven’t looked at either of MacDonald’s books)
If you’re interested in geometric modeling for robotics, computer graphics, computer vision, or similar, check out [these papers][invkinematics] and then look at the book [Geometric Algebra for Computer Science][gacs]. The “conformal geometric algebra” model proposed there is pretty neat. Also see [these papers][unifalg] about related topics.
If you’re interested in crystallography, check out the papers [“Point Groups and Space Groups in Geometric Algebra”][crystalsymmetry] and [“The Crystallographic Space Groups in Geometric Algebra”][crystalga].
If you’re interested in Lie theory or representation theory, check out this paper [“Lie Groups as Spin Groups”][lgasg].
If you’re a physicist / physics student / electrical engineer / etc., try the book [Geometric Algebra for Physicists][gap] or perhaps [Understanding Geometric Algebra for Electromagnetic Theory][gaet]
For a slightly different perspective, take a look at [Sobczyk][]’s [New Foundations in Mathematics: The Geometric Concept of Number][nfm]. Sobczyk has some interesting papers about representing geometric algebras using (real or complex-valued) matrices, which might be helpful if you want to write fast numerical code on current computers, which have been optimized to do matrix math.
If you’re interested in the history, I recommend Crowe’s 1967 [History of Vector Analysis][hva].
I believe you can find the collected mathematical papers of Clifford online if you do a google search, or buy a used copy of a nice version published by Chelsea in the 1960s; the recent AMS reprint is awful quality.
Grassmann’s two mid-19th century Ausdehnungslehre books have been relatively recently translated into English, as has Peano’s late-19th century book Geometic Calculus, all three by Kannenberg.
[oersted]: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
[gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf
[chisolm]: https://arxiv.org/abs/1205.5935
[nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html
[planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab....
[asu]: http://geocalc.clas.asu.edu/
[cambridge]: http://geometry.mrao.cam.ac.uk/
[netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html
[lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/
[counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl...
[MacDonald]: http://faculty.luther.edu/~macdonal/
[laga]: http://faculty.luther.edu/~macdonal/laga/index.html
[vagc]: http://faculty.luther.edu/~macdonal/vagc/
[cagc]: http://geocalc.clas.asu.edu/html/CA_to_GC.html
[invkinematics]: http://geocalc.clas.asu.edu/html/InvariantKinematics.html
[gacs]: http://www.geometricalgebra.net
[unifalg]: http://geocalc.clas.asu.edu/html/UAFCG.html
[crystalsymmetry]: http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf
[crystalga]: http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf
[lgasg]: http://geocalc.clas.asu.edu/pdf/LGasSG.pdf
[gap]: http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for...
[gaet]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-047094163...
[Sobczyk]: http://www.garretstar.com
[nfm]: http://www.springer.com/us/book/9780817683849
[hva]: https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis
Alan Bromborsky, An Introduction to Geometric Algebra and Calculus http://www2.montgomerycollege.edu/departments/planet/planet/...
Also some more links at http://www2.montgomerycollege.edu/departments/planet/planet/...