I suggest the short book by Robert, "Introduction to the Representation Theory of Compact and Locally Compact Groups" which is leisurely and has plenty of exercises. The only prerequisite of this book is some familiarity of finite dimensional representations.
A second book you should look at is Folland's "A Course in Abstract Harmonic Analysis", which is more advanced, and requires more experience with analysis (having seen Banach spaces is not a bad thing), but the advantage of this book is that it has very clearly written proofs, that are easily to follow (I do algebra mostly, and I find many analysis tracts a bit opaque in this regard). Unfortunately, this book does not have exercises, and should be approached once you have plenty of examples in mind.
Donald Cohn's "measure theory" has a large number of exercises on the basics of topological groups and Haar measure, but it doesn't do representation theory or much else on locally compact groups except an introduction.