You could do worse than just getting a good thick biology textbook (I have Campbell & Reece). The bits that aren't specifically about evolution are also interesting and put the whole thing into context. You can pick up such books relatively cheap second hand, and usually in good condition.
As far as the Big Bang goes, I fear that it isn't really possible to do cosmology without learning quantum theory and General Relativity --- analogies about rubber sheets can only take one so far. I suppose as far as popularizations go,
A Brief History Of Time is better than most --- at least it was written by a physicist rather than a popular science writer.
Edited by Dr Adequate, : No reason given.