Polya & Velleman are classics. They're about problem solving & proof writing techniques, but won't teach you any abstract math. I've only ever heard good things but i've actually never read them myself! Some other suggestions:

Niven - The Theory of Numbers. Contains hints for some of the proof based exercises, and answers for many of the computational exercises. I used it for my undergrad course, I remember it being reasonably beginner friendly.

Pressley - Elementary Differential Geometry. Has terse answers to every exercise! The subject is a nice mixture of concrete and abstract, calculation and proofs, and there's some interesting work using differential geometry in CS via computational geometry. Also i think it's used in robotics and a few other things? This book is very beginner friendly, it will get you about 1/2 way to General Relativity.