I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I've had is computing things like the join, product, coproduct, pullback, pushout, and so forth. I understand them as far as their universal properties, and maybe have a little intuition because the category of simplicial sets is a presheaf category over the simplex category, but Lurie uses geometrical language, so I can't even compute like when working with presheaves. So, could you lot recommend some books or lecture notes, *preferably with suggested sections*, that won't go too deep into simplicial homotopy theory, but deep enough for me to learn how to compute?


Note: Most of the time I waste is sitting around with that book is trying to make sense of the computations, while I understand the arguments just fine. So please, don't suggest an entire book detailing all of simplicial homotopy theory from start to finish. I have a goal in mind here, and I'm only trying to learn as much as is necessary for me to continue reading HTT.



Simplicial homotopy book suggestion for HTT computations
I'm not really sure what you are asking for. Colimits and limits are easy to compute in simplicial sets, because it's a presheaf category (as you say).
But if you want "geometrical" intuition about simplicial sets (including "pictures" of joins, etc.), you want to know about the geometric realization functor from simplicial sets to spaces. In particular, the fact that geometric realization preserves colimits (it's a left adjoint), and that it preserves finite limits (essentially a theorem of Milnor). 


The limit result is proved in the book by Gabriel-Zisman. (Goerss-Jardine mention the result in chapter 1, but don't seem to prove it.)


