start with Thompson's book. Then Ebeling on lattices, which helped me a good deal. Let me look up full titles
Thomas M. Thompson, From Error Correcting Codes through Sphere Packings to Simple Groups
Wolfgang Ebeling, Lattices and Codes there is now a third edition
Well, why not. A technique that was probably invented by Conway shows that any "even" lattice with covering radius strictly below $\sqrt 2$ has class number one. G. Nebe first published a list of these, maximum dimension possible is ten (goes back to G. L. Watson). I and Pete Clark published a small correction by finding lattices with class number one. Meanwhile, the Leech lattice has covering radius exactly $\sqrt 2,$ and class number $24.$ The thing about the covering radius is that one may draw pictures for the two dimensional case, on ordinary graph paper, and learn a good deal.