I haven't read it myself but I know that the Russian original is based on material taught in the third part of an analysis course at the top national university. No idea why the English edition was titled "Introduction".
Well, trigonometry is the only one which is a bit like a textbook. It is still kind of short (but I guess the length is normal for a school textbook). Of course, Gelfand being one of the top mathematicians, one can still learn a lot from anything he writes. Same with Polya. I just wouldn't...
Everything by Lang in general is very good but can/should be supplemented for additional reading. If your goal is to read Spivak, why not read a standard calculus textbook first? I liked Lang's "A First Course" very much. It's a bit lengthy though, so you'll probably want to skip some parts...
If you want a middle ground between Spivak and Rudin, then Bartle and Sherbert's Introduction... is quite reader-friendly. It is not as much fun as Spivak, but it does the job in a systematic manner starting from foundations. It is also a proper first real analysis book.
I am currently reading Statistics by Freedman, Pisani and Purves. It is fantastic. The book teaches how to think about statistical methods and concepts deeply while keeping the mathematical level very accessible. For this reason one will need to read more advanced material later, but a very...
It is completely possible to fluke GRE quant even when you feel confident in you math skills. I myself got very bad scores once due to stress and lack of sleep (and also not preparing for it seriously). When I retook it later, I had 790 (99th percentile) score without much trouble. You should...
I like Elementary probability theory by Kai Chung. You may want to supplement it with chapters from a discrete mathematics book for combinatorics. The other option would be reading a more advanced book to get the intuition, and then coming back to the level of Ross for problems.
K&R can be read by anyone still in high school. It is very well written, accessible and short.
As for the above recommendation to start with C++, it's a terrible suggestion.