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).