I'll write here a list of my personal favorites plus some commonly used books.
I wouldn't be surprised if your teacher chose either one of the books below as a textbook:
i) Mechanics, the first volume of the Landau course on Theoretical Physics;
ii) Goldstein's book "Classical Mechanics";
iii) Taylor's book "Classical Mechanics";
iv) Marion's book "Classical Dynamics of Particles and Systems";
v) Symon's book "Mechanics";
Goldstein's book may be very appropriate for a first or second course on the topic, but I don't believe it displays a very formal approach to the subject. I'd suggest it to someone who's not interested in the mathematical structure of Mechanics. Even though, good for a starter.
Taylor's book has some very good exercises, but the book itself does not please me at all since it's informal, prolix and severely incomplete in most topics. Same goes to Marion's book, and even though Symon's is a little bit better, it didn't please me either.
The best book in this list if definitely Landau's, but I don't find it as good as most people picture it. I didn't read the whole Landau series (not even half, actually), but until now it's the worst of them all, for me. It still carries much of the author's incredible insights and some very nice solved exercises, but (as Arnol'd pointed out) there are a some mistakes and fake demonstrations on the book. Don't trust all of his "proofs" and you'll be safe.
Now I'll point some books that really helped me throughout my studies:
Arnol'd's "Mathematical Methods of Classical Mechanics":
This book is simply the best book you can get your hands on after acquiring familiarity with the subject (after a first course using Goldstein's or Landau's book, for example). It's thorough, the maths are just clear and not extravagant, the proofs are very simple and you can get some contact with phase space structures, Lie algebras, differential geometry, exterior algebra and perturbation methods. Arnol'd's way of writing is incredibly clean, as if he really wanted to write a book with no "mysteries" and "conclusions that jump out of nowhere". The exercises are not very suited for a course.
Saletan's "Classical Dynamics: a Contemporary Approach":
Very nice book. A little more developed mathematically than Arnol'd's, since it delves into the structure of the cotangent bundle and spends a great deal of the book talking about chaos and Hamilton-Jacobi theory. The proofs are not very elegant, but I'd chose it as a textbook for a graduate course. Some nice exercises.
Fasano's "Analytical Dynamics":
Also a graduate-textbook-style one. Very close to Saletan's way of writing, trying to explain to physicists the mathematical nature of Mechanics without too much rigor, but developing proofs of many theorems. Very nice chapter of angular momentum, very nice exercises (some of them, solved!). Incredibly nice introduction to Lie derivatives and canonical transformations, and very philosophically inclined chapters so to answer "why is this this way" or "what does that mean, really?".
Lanczos' "The Variational Principles of Mechanics":
This book is kept close at all times. Not suited (at all) as a textbook, more like a companion throughout life. The most philosophical, inquiring and historical Mechanics book ever written. If you want to read a very beautiful account on the the structure, the problems, the development and the birth of mechanical concepts I'd recommend this book without blinking. It is a physics book: calculus and stuff, but looks like it were written by someone who liked to ask deep questions of the kind "why do we use this instead of this, and why is mathematics such a perfect language for physics?". It's just amazing.
Marsden's "Foundations of Mechanics":
This is the bible of Mechanics. Since it's a bible, no one ever read it all or understood it all. Not to be used as textbook ever. It's a book aimed for mathematicians, but the mathematical physicist will learn a lot from it, since it's quite self contained in what touches the maths: they're all developed in the first two chapters. Even though, very acidly developed. Hard to read, hard to understand, hard to grasp some proofs... In general, hard to use. Even though, I really like some parts of if... A lot.
Ana Cannas' "Introduction to Symplectic and Hamiltonian Geometry":
Another mathematics book, but this is the best one (in my humble opinion). Can be found for free (in English) at www.impa.br/opencms/pt/biblioteca/pm/PM_11.pdf .
Kotkin's "Collection of Problems in Classical Mechanics":
Last but not least, filling in the "with a lot of exercises" hole, Serbo & Kotkin's book is simply the key to score 101 out of 100 in any Mechanics exam. Hundreds of incredible, beautiful, well thought problems together with all (ALL!) their solutions at the end. From very simple to "hell no I'm not trying this one" problems, this book should be a reference to everyone studying the subject. Some of the problems are so nice that you can even publish notes in teaching journals about them, like I've seen once of twice before.
Well, this is my humble contribution. I hope it helps you!
EDIT.: I just noticed I forgot one book that really changed my life: Spivak's "Physics for Mathematicians, Volume I: Mechanics". The physicist should not be scared about the title. This is the best book ever written about Mechanics. I actually have plans of taking vacations only to read it all. There's nothing missing, all the mathematics is rigorous and perfect, and there's not a single step that isn't clarified by the author (who said he was learning Mechanics himself whilst writing this book). There are moments he pauses to inquire about contact structures in symplectic manifolds, but also moments where the reason for inquiry is the fact that forces are represented by vectors; and then he goes back to Newton's time where vectors didn't exist... And tries to explain how people used to see forces and momentum at the time, in his opinion. It's just magical. He's as worried about presenting the content of the subject as to try to grasp why the definitions are the way they are, and then justify it historically. Sorry if I'm being redundant, but please read this book!