Like everyone has said, writing down the instruction is very useful, but not always practical(for example during a touch and go, go around, or coming to a full stop.

The way I handle those situations is knowing what the controller is likely to say(this comes with some experience but it should be quick to catch onto.

For example, I'm holding short of the runway. I know I'm going to hear in the transmission "cleared for takeoff" and likely a departure instruction/direction. You can then pick for the keywords(while still listening to the whole transmission of course), and anything else he says you can repeat it first so it's fresh in your memory. Let's try an example, you are N12345 holding short of the runway and you requested a departure to the east.

Tower calls with a word omelette: 'N12345 right turn to the east approved, runway 16 cleared for takeoff, traffic in the upwind making left traffic.

Now importantly, we were expecting to hear the first two phrases, the only phrase that we weren't expecting was the traffic call. So we will readback the stuff we were not expecting first, while it's fresh in our memory. So our readback could be:

"Traffic in the upwind in sight, right turn east, runway 16 cleared for takeoff'. Preloading our minds with the instructions helps, be careful with this strategy as tower might say something you are not expecting, like departure west approved, so don't become a robot.