Pilot reported they descended below their assigned altitude and received a Low Altitude Alert from ATC.

Narrative

On Day 0 I was conducting a flight from ZZZ to CHA. I just recently transition from my Aircraft Y to the Aircraft X. The major difference between the two is the Garmin G3000 integrated flight system. I'm still learning all the features and it's a very comprehensive system. I felt it was good to do some practice flights in real world situations so I filed multiple flight legs; to gain more experience with the G3000. It was on the second leg going to CHA with Chattanooga Approach when the incident happened. It was getting quite busy with both the controller and myself in the cockpit. I was given a clearance to descend to 3000 and fly heading 290. I responded by saying '290 Aircraft X'. The controller didn't pick up the fact I didn't read back the assigned altitude and he moved on to other traffic. I then inadvertently set 2000 in the altitude preselect. The controller then got caught up working an Aircraft Z pilot and by the time he got back to me; I was given instructions to fly 230 and join the approach course. The controller was so delayed that by the time I turned the heading bug to 230 and armed the Nav feature; we had already flown through the final approach course and as a result the autopilot never captured the approach course and we continued to fly north of the final. It was soon after the controller advised a low altitude alert and instructed me to climb to 2700 feet. On landing I was given a call by the Controller Supervisor to call him regarding a possible pilot deviation. I feel there were multiple mistakes made both on my part and on the controller. I didn't read back an altitude clearance; and the controller didn't respond correctly by confirming my altitude read back. Secondly controller was late in giving me instructions to join the final which resulted in not capturing the final and getting lower than 3000 feet which caused the low altitude alert.