Mechanics: From Newton's Laws to Deterministic Chaos

Although I agree that one might be scared by the presence of sophisticated chapters like the one devoted to differential geometry, I find that this book is the best for an undergraduate level. It begins with Newtonian mechanics (some knowledge on linear algebra and calculus assumed) and covers all the fundamental points of modern mechanics in a rigorous and straightforward way, including a chapter in special relativity and another on dynamical systems and chaos. As it emphasizes the relevant role of symmetries, it naturally leads to quantum mechanics and can be used by advanced undergrad or graduate students interested in the geometrical foundations of mechanics.
Even though it is certainly not the easiest book available, if you have to buy just one, buy this!