Maybe you will find useful the notes by Sjamaar - "Manifolds and Differential Forms" which can be downloaded for free at his website. The explanation is geometrically motivated and straightforward from the ground up, and it contains lots of doable exercises and explicit detailed examples which may help you grasp everything you need to know and more. Donu Arapura has a nice elementary summary of the concepts and uses of differential forms in his notes Arapura - "Introduction to differential forms" freely downloadable too.
The most explicit, introductory but detailed, and full of exercises references for an elementary introduction to all of this, are the books:
- Weintraub - Differential Forms, A Complement to Vector Calculus.
- Bachman - A Geometric Approach to Differential Forms.
(The second one has a draft old version available online, but the second edition of the book has been very improved).
As a conceptual complement, a very interesting book geared toward theoretical physics applications is Baez/Muniain - "Gauge Fields, Knots and Gravity" where the meaning and extensive use of covectors and differential forms in general is used as a primary tool to formulate physical theories in geometric terms.