Well, how can I resist? My Introduction to Gödel's Theorems (CUP, 2007; second edition forthcoming in March 2013) was written precisely for those with little prior background in mathematical logic, and aims to be very accessible while actually giving the main proofs in quite a bit of detail.
Other options that range a bit more widely into logic and the theory of computation include George Boolos and Richard C. Jeffrey, Computability and Logic (CUP, 3rd edn 1989: there are expanded later editions with John Burgess as a third author, but these are arguably less elegantly done). Richard L. Epstein and Walter Carnielli’s very nice Computability also discusses
computation in general before covering the incompleteness theorems; this
is attractively written with a lot of interesting historical asides.