Is there an easy way to remember all the basic graphical transformations?

Well yes! Say you're given an equation of y=f(x), you begin by sketching the graph if plotting it is not part of the question, the reason you do this is to get an idea of what the graph looks like (turning points & roots can be visualised). Next you need to understand how each operation affects the original graph, for that all you need to remember is that any operation done within the bracket, for example y= f(x-3), affects only the x co-ordinates of the graph. And any operation done outside the bracket, for example y= f(x)-3, affects only the y co-ordinates. And to remember what each operator does, all you need to remember to do the exact opposite of the operator for x. And for y you do exactly what it says.

So going back to our examples, y= f(x-3), the operator -3 is applied to x so all you do is the exact opposite when plotting the graph, you add 3 to all your x co-ordinates. And for y= f(x)-3 , the operator -3 is applied to y therefore you need to do exact operator for all your y values by taking away 3.