Introductory books as preparation to read Voevodsky homotopy-theory (HoTT) book

Solution 1:

You can take a look at Robert Harper's lectures: http://www.cs.cmu.edu/~rwh/courses/hott/ (There are lecture notes and video recordings) They require much less than you described to understand and covert HoTT.

Solution 2:

Another way to look at those notions of groupoids and fibrations is that type theory tells you what they are. It does so by telling you what you can do with them.

For example with fibrations, you can transport points in the fibers along paths -- this is exactly the intention of the classical notion of fibrations. It is even more true for weak $\infty$-groupoids.

So instead of reading lots of 'background' material which explains you the classical versions of the terms that are used in the HoTT book, you maiy find it easier and quicker to learn what they are through the explanations that the book provides.

If you don't feel comfortable with the idea of learning what things are by learning what you can do with them, I'd suggest with starting to read about homotopy theory itself. Hatcher's Algebraic Topology is probably a reasonable starting point (which is freely available online).