It seems to me that Rudin's book should be the easiest for you since you already know a fair amount of calculus from a fairly abstract treatment. You ought to make the fastest progress with Rudin's book. Linear Algebra interacts with analysis really well. In quite a few fields which have use of applied mathematics, a strong foundation in linear algebra and analysis constitute the bulk of mathematics people know and use regularly. So, this should set you up to be able to read quite a bit of physics, engineering, machine learning etc should you so choose.
Algebra is an entirely different direction. At least, in my experience, it's likely to feel that way. Quite a few people, who have a head for analysis, flounder in algebra; and vice versa. This might not be you, but it's something to keep in mind.
So, my best guess as to order of ease: Analysis, Linear Algebra, Algebra. My best guess as to what will be most useful to you if you are an applied person: Linear Algebra (since you already know a lot of analysis), Analysis, Algebra. In order of what might be least like the background you've mentioned: Algebra, Linear Algebra, Analysis.
I don't know enough to tell you what to do directly, but hopefully my answer has provided enough guidance for you to make the decision for yourself.
(Have you considered a more concrete Linear Algebra book like Strang's "Linear Algebra and Its Applications"? It seems like you are starting at few levels up of abstraction. I find it's good to get a sense of how the calculations work at a lower level first.)