Here’s a comment I made elsewhere recently:

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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