"Foundations of Computer Science" by Aho and Ullman might fit your bill. Its focus is more on the concepts and principles of computing rather than math-centric CS theory.
http://infolab.stanford.edu/~ullman/focs.htmlAnother nontraditional intro to computing worth mentioning is "Structure and Interpretation of Computer Programs" by Abelson and Sussman(s). It teaches programming concepts using the Lisp/Scheme language (seldom used any more), but does so brilliantly.
"Think Python: How to Think Like a Computer Scientist" by Downey might provide the right mix of computing concepts and programming practice.
If you do want a traditional intro to CS theory, two books that cover that topic well are: "Introduction to the Theory of Computation" by Sipser, and "Introduction to Automata Theory, Languages, and Computation" by Hopcroft, Motwani, and Ullman.
Three very good books that introduce algorithms are: "Introduction to Algorithms: A Creative Approach" by Manber, "The Algorithm Design Manual" by Skiena, and "Algorithms" by Sedgewick and Wayne.
Remember that CS theory doesn't age, so buying a used early edition of a textbook should serve your needs just as well as an new up-to-date edition.