Make sure to keep track of every object in a question. If you're trying to find the momentum of two cars after a crash, remember that both cars are probably moving after the crash, not just the one that got hit.
Being able to break a problem into sections and define isolated systems is key to solving problems in physics—but be careful not to forget the parts of the question you've temporarily ignored. Always check to make sure every force in the question is accounted for.
It's tricky to get the hang of negative potential energy. Just remember that since potential energy is a relative measurement, we're allowed to start counting from anywhere and call it zero—as long as we're consistent in where we say zero is. You can only set h = 0 once, not at every step of a problem.
When you pick a point in time to analyze, be sure to capture all the forms of energy at that instant—and no more. Some kinds of energy may appear and disappear as an object moves (a snowboarder in a half pipe has potential energy, then kinetic, then potential again), but the only energy that matters is what's present at the exact instant you chose to examine.
If an object is stationary, all of these formulas still apply—a stationary object just has a velocity of (you guessed it) zero.
When you calculate total momentum, stationary objects contribute zero, but can gain momentum if they start moving. Velocity is directional, so sometimes v1 or v2 will be negative, and in order to calculate total momentum you must subtract. Two identical cars driving away from each other at the same speed, for example, have a net momentum of zero.
Morals of the Icarus story: 1) you gotta make your wings out of carbon fiber, not wax, bro, and 2) pay attention to how outside forces can change the form of energy of an object. Don't forget that forces like friction are going to remove mechanical energy from an object, while other outside forces can add mechanical energy in the form of either kinetic or potential energy.
The formula P = Fv is straightforward, but you have to remember that the object expending the power to make the force doesn't have to be the object moving at speed v. Two examples: in a car, the engine combusts gasoline to generate power that makes a force to move the car forward; the more horsepower*, the faster the car can move. But a hang glider uses force from the wind to move (so F would come from the wind) and v from the glider, together giving us the power the wind uses to move the glider.
*1 hp = 746 W