I got really stuck to the end of Guillemin and Pollack (in particular, here) and plan to give up.
Give up Guillemin and Pollack, not math though.
It seems John Milnor's classic little book topology from the differential view point is an ideal alternate for the first three chapters. So I am thinking of starting that one - good idea? If so, how about the forth chapter on calculus and cohomology in Guillemin and Pollack? What would be an more comprehensible alternative? A reference of a book directing to the specific chapter(s) would be really appreciated.
And also I can just restart Guillemin and Pollack.
So I am wondering the best move at the moment?
My advice (which is something generic to life) is that things do get easier. You might struggle with a particular topic in mathematics for a while but if you keep working at it, then eventually you will improve and your previous mathematical self will be a shadow of your current mathematical self. Of course, sometimes it's just not productive to read a particular textbook anymore and it's better to switch references.
I think as Ryan Budney hints at, it might not be the actual differential topology that is challenging but rather the underlying concepts from multivariable calculus. Do you think that would be an accurate description of things? If so, then one thing that's advisable to do is to go to the bare minimal definitions of what you're reading, e.g., differential forms, pullbacks, the de Rham differential, integration on manifolds. You might re-read this material but when you see a new definition, make it a habit to compute something with it. I can see that's what you're doing based on your questions here and it's excellent. The wonderful thing is that whenever you're stuck, you can just ask a question here.
Milnor's book is a condensed treatment (roughly 60 pages?) of differential topology. However, one book that you might like is Differential Forms in Algebraic Topology by Bott and Tu. It's an excellent textbook that gives a different perspective of differential forms via their use in algebraic topology. You might find it interesting.
I hope this helps!