First, I'd get a copy of the service (not operator) manual for your make/model of motor.
Second, I'd get a book coverign marine electrical. Chapman's has a chapter on marine electrical, and is a good start point, but doesn't have the depth necessary to diagnose and fix many things.
A good basic electricity book that covers AC and DC circuits and how to use a multi-meter to discover shorts, open circuits, excessive resistance at connectors, etc. would also be benificial.
BTW, it sounds like your problem is wiring. Did the battery have enough juice to crank the motor? If so, then there is a bad connector, broken wire, or fuse popped. If not, then first check the battery connections to make sure you are getting a good connection between the battery posts and the wires. A bad connection will keep the battery from getting a good charge, and will increase resistance to the point that large current draw motors (starting and tilt/trim) won't run. Clean the battery posts and all wires that attach to the battery, make sure the battery is charged and try again. If that doesn't solve the problem, then it is a wire/connector/fuse in the tilt/trim circuit...or possibly the tilt/trim motor itself. But before replacing a motor, absolutely rule out a battery or wiring problem.
Before replacing a battery:
1. Make sure it is topped off with distilled water (yes...even the so called "maintenance free" batteries are not truly maintenance free and need water from time to time)
2. Make sure all the connectors are clean and bright
3. Make sure it is charged and holds a charge
4. Make sure the alternator on the motor is actually charging the battery (no use killing a brand new battery to discover this)
Before replacing the tilt/trim motor:
1. Rule out a battery problem
2. Rule out a switch problem
3. Rule out a fuse/circuit breaker problem
4. Rule out broken wires, high resistance at connectors between the tilt/trim motor and the tilt/trim switches and between the tilt/trim switches and the battery.