Seriously, that's not too far off. Notch filters happen to be particularly difficult to get the math right if you're making one from scratch, by the way.
It depends on what you're trying to do. Generally, when you build a 2nd- order filter (or a higher-order filter from 2nd-order sections), you use one of the common "direct form" architectures. There's advantages to each -- and danged if it hasn't been long enough since I've needed to make one that I can't remember!!
Q or damping ratio for 2nd-order filters.
Passband ripple, stopband ripple, and shape factor (or transition band steepness) for higher-order filters.
Often neglected, but essential, is the degree to which quantization noise gets amplified (there's a whole direct-form 1 vs. direct-form 2 spiel out there on just this subject).
Rick Lyons' "Understanding Digital Signal Processing" is probably best if you have just one book to go on. I feel that it doesn't quite get the beginner all the way up from nothing to understanding -- but there's no one book that does, to my knowledge.
If you were interested in control systems I'd direct you to my book. In fact, where Rick's book glosses over the z transform, I think mine does better. Oppenheim & Wilski's "Signal Processing" is the de-facto standard book, but you may find it rough going without taking a class to go with it (for that matter, you may find it rough going _with_ a class).
Do you still have the book?
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.