My thoughts as a math major about to graduate:
If you don't remember your high school math, I'd do the khan academy courses on pre algebra and algebra. A textbook would probably be overkill.
Start with Book of Proof by Hammack. It's important to learn basic proof writing (and reading) techniques before you dive into more complicated math. The book goes into set theory as well, which is also really important to being able to understand pretty much every field of math. I'd skip the chapters on combinatorics unless that stuff really gets you going.
From here, I think that it's actually be good to learn some topology. Topology is really useful in a lot of areas of math, and I think that analysis will make more sense if you learn some topology first. Topology by Munkres is a good reference, but I personally really liked Basic Topology by Armstrong. It's way more readable, but having Munkres around for more rigorous definitions and examples will probably be helpful.
Next, you can learn analysis. Analysis 1 by Terrence Tao is supposed to be pretty good, but I haven't read it personally. Real analysis will hopefully make more sense with some knowledge of topology. For complex analysis, I like Complex Variables by Fisher. If you want something more verbose, there's Complex Variables and Applications by Brown and Churchill.
For learning algebra, there's only one book in my opinion, Abstract Algebra by Dummit and Foote. This book is dense, but is has everything that you'll need relating to algebra for all of undergrad (as well as some graduate topics). Parts 1 and 2 cover what would probably be a one semester course on algebra, and part 3 covers linear algebra. The linear algebra section might be a bit hard to get through, but unfortunately I haven't read any other linear algebra books so I don't have any recommendations.
That pretty much covers it for an undergraduate curriculum!
