The notation really isn't as bad as it originally looks, trust me. I was in the same boat recently, but after cracking open TAPL, on page 26 you basically get all the notation you need for the rest of the book.

Inference rules just transform the traditional format of

    (A && B) -> C

into

    A B
    ---
     C

This format allows you to stack rules on top of each other when writing out the derivation of some term, without running out of page space.