Here are a few suggestions (which depending on your background may be more or less useful):
- Logic and Structure by Dirk van Dalen. I have used this as a textbook when teaching mathematical logic and for that purpose it is decent. Some people find it a bit dry, but at least it covers a large amount of material in a reasonably clear manner.
- Mathematical Logic by Joseph R. Shoenfield. This book is, I think, regarded by many logicians as being the gold standard text on the subject.
- A Course in Mathematical Logic by John Bell and Moshe Machover. This is my personal favorite textbook in mathematical logic. (Unfortunately, it's a North Holland book and so is a bit less affordable.)
- A Course in Mathematical Logic for Mathematicians by Yuri I. Manin (with contributions from Boris Zilber). I think that pretty much anything written by Manin is worth taking seriously and this book is no exception.
- Notes on Logic and Set Theory by Peter T. Johnstone. This is a delightful little (literally) book on logic which is highly recommended (perhaps in conjunction with one of the other larger books from this list).
- The Mathematics of Metamathematics by Helena Rasiowa and Roman Sikorski. This is a nice book which gives a lattice theoretic development of mathematical logic. (Difficult to find, but worth a look if your library has a copy.)
- Introduction to Metamathematics by Stephen C. Kleene. A classic text in mathematical logic which is still a rewarding read.
I hope these (admittedly biased) suggestions are some use!
