I really liked "Riemannian Geometry" by Manfredo do Carmo - http://www.amazon.com/dp/0817634908/ref=rdr_ext_sb_ti_sims_1 (a PDF copy can be found by googling "Riemannian geometry do Carmo" and looking at the first few search results)
Chapter 0 discusses the preliminaries from smooth manifold theory and subsequent chapters immediately begin in Riemannian geometry (Chapter 0 is only roughly 30 pages in length and the entire book is roughly 300 pages in length). Of course, one should be warned that Chapter 0 is quite terse and I think it is better to have some familiarity with smooth manifolds beforehand. However, with enough mathematical maturity, it should be possible to learn Riemannian geometry from do Carmo without any background in smooth manifold theory, beginning with Chapter 0.
Also, another alternative is "An Introduction to Differentiable Manifolds and Riemannian Geometry" by William Boothby - http://www.amazon.com/Introduction-Differentiable-Manifolds-Riemannian-Mathematics/dp/0121160513
I really liked this book - it covers both smooth manifold theory (at roughly the level of Lee but in the space of 300, rather than 500 pages) and also covers Riemannian geometry in two chapters. The depth of coverage in Riemannian geometry is not very much but the coverage of smooth manifold theory is quite efficient compared to Lee. You could just read chapters 1 - 5 (roughly 200 pages with relatively large font) and skip chapter 6 (chapters 7 and 8 are on introductory Riemannian geometry, which you can read if you like, or move onto a more specialised textbook in the subject such as Peterson or do Carmo).
Take a look and let me know what you think!