During my undergrad, we had a reading assignment (for most courses) which would nearly cover up the entire textbook including introductions, summaries, "did you know" and other such fluff material.
In graduate school as pointed out in most answers here on SE, one does not read the whole book cover to cover but just read the parts you need and backtrack if doubts. But is this also true if I'm starting out in a new field?
I had my BS in Engineering and I'm pursuing my grad studies in Math, certain topics like Topology are completely new to me. I don't need the whole of Topology but just certain bits and pieces.
Should I attain some familiarity with the topic by reading a good introductory book cover to cover or just dive in (into a completely alien field) and understand only the parts I need?
I think this has been covered here in prior questions, but to continue on anyway ...
You should read material with a purpose or a goal. If the material is helping to achieve that goal (learning new or foreign material that you have a desire to learn) then reading all of the material is fine. There is no point in reading material though if it has no purpose or a goal (e.g. topics you are not interested in, material you already know sufficiently).
Personally for me it is a mix, some material I skim in large parts, some material I read only portions that I'm directly interested in, and if I'm really engrossed in some material I will read it front to back. Currently I'm reading an introduction cartography text book and really enjoying it and reading every chapter! It would be torture for me though to read though an intro textbook for criminal justice, criminology or sociology.
Same goes for journal articles, posts on Q/A forums, etc.