There is relatively limited number of things that a controller will be saying to you as a pilot, and they're generally grouped by the phase of flight. Once you learn the various key phases and phrases, things become MUCH easier.
Here's an example of just how important 'context' can be. I was transitioning overhead Teterboro on a VFR sightseeing flight towards NYC. The tower controller said, "say parking." I couldn't hear him very well, and since I thought it was understood that I was transitioning, not landing, I couldn't for the life of me work out what he was saying. I had to have him repeat it a number of times before I made out that it was "say parking."
An IFR precision approach clearance with vectors to final goes something like this, "N123AB, 5 from the marker, fly hdg 220, maintain 2500 until established on the localizer, cleared ILS rwy 25R approach." To the uninitiated, you truly have no PRAYER of following that, let alone repeating it back. Once you understand that ATC is responsible for giving you your position, and an altitude to maintain until being established on a segment of the approach when you're on a vectored/unpublished route, then it starts to make sense.
Even without that specific knowledge, you can simply take note of the fact that vectored approach clearances generally sound like that, and that the heading and altitude are "main" items to pick up. Realizing that ATC will have you intercept at a 30 deg angle or less, the heading shouldn't come as news, either. Overall, the ONLY piece of 'news' in that clearance is the altitude, and even that will often be predictable. The callsign, position, heading and the approach clearance itself should be fully expected by the pilot.
While you're VFR enroute, calls for you are likely to be traffic, altimeter settings, or verification of your destination. Once you're approaching the terminal area, you'll start being asked to report the field in sight. With that done, once you're handed off to tower, expect either a landing clearance, and pattern entry instruction, or notification of where other traffic is in the pattern.
It goes on, and on. Understand the calls associated with each phase of flight (under VFR for now, and then under IFR if you pursue that later on), then when you hear them over a crackly radio, your brain will be able to associate it with a standard library of instructions, rather than literally trying to hear every word and 'start from scratch' in your attempt to process it.
I hope this helps.