I have a background in Analysis, specifically with Baby Rudin. However, as many people note, Rudin does not do a very good job discussing differential forms. Could someone please refer me to an online resource which will help me best understand differential forms with the background I already have?
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There are similar questions on this site, you should search and find more materials referenced there, besides those I give here. For example there are good free online notes by Sjamaar - "Manifolds and Differential Forms" which require almost only advanced vector calculus and linear algebra to be studied. Another nice elementary summary of the concepts and uses of differential forms are the notes by Arapura - "Introduction to differential forms". Finally, there is a draft old version available online of the popular book Bachman - A Geometric Approach to Differential Forms.
I recommend From Calculus to Cohomology by Madsen and Tornehave. It covers differential forms in detail, both in Euclidean space and on manifolds, and gives an introduction to de Rham cohomology, the main tool for studying them. You can also use it as a first exposure to algebraic topology.
I was recently in your situation and encountered this book
It is written at a beginner level and is very easy to read for some with your background. Plus, it combines all of the relevant linear algebra and vector calculus into one book leading up to differential forms.
Another book that is extremely well written which covers the topic is Loring Tu's "An Introduction to Manifolds."