Which one should I take first? Does it help to take one before the other?
Depends on your interests.Which one should I take first?
The two subject matters are largely unrelated. As long as you are comfortable with logical thinking and proofs as in the normal upper division linear algebra you should be fine.Does it help to take one before the other?
Not really. But I am majoring in applied mathematics. So I guess the question is: which would be more useful for my major?Depends on your interests.
Have you developed any interests yet?
For Analysis and Algebra, it would be helpful to know basic set theory, 1-1 functions, etc. I assume most of it will be covered in the first few weeks.Also, do I need to know anything for real analysis or algebra besides proof skills? I know that my school has prereqs, but sometimes not everything is used.
Dummit and Foote is probably the easiest book to work from, but the best intro is probably Michael Artin's book.I studied algebra after analysis, so had built some mathematical maturity by then. I found a lot of the introductory algebra books to be boring and not very challenging, e.g. Gallian. The course I took nominally used Dummit and Foote, but the professor followed his own notes and I hardly ever looked at the book as it seemed very longwinded and dry. I read part of Herstein's "Topics in Algebra" instead, and really liked it, particularly the group theory parts. That book would be my first recommendation to anyone new to the subject but not new to proof-oriented mathematics in general.