A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT should be skipped (example guide).
This question asks for a similar guide for learning algebra in the context of $(\infty,1)$-categories, at the level of generality of Lurie's Higher Algebra.
More specifically, the focus should not be on DG-algebras, being instead about more general mathematical objects such as $\mathbb{E}_k$-rings.
Nice things to include in such a guide would be other helpful sources, parts that should be skipped, or concepts that are best treated as black boxes (at least when first approaching it).
Here is the sister question to this one, which asks for a roadmap to Lurie's Spectral Algebraic Geometry.
Best Answer
I think that reading the proof of straightening and unstraightening is probably a great way to get bogged down in the details and should be treated as a black box unless you interested in doing 'pure' higher category theory. The proof is nontrivial and also somewhat unenlightening (the reason that it works, especially in the marked case, involves some intuition that Lurie had about lax cones in $(\infty,2)$-categories that is not at all clear from the exposition, where things appear as if by magic).
I would also highly suggest Cisinski's new book as an introduction before reading HTT and HA. Some of the proofs in HTT are dated and can be done much more easily (not yet for marked straightening, though).