Before you can tow you need a hitch. See it you can find one for your Civic and what the cost is installed.
The automatic transmission in the Civic isn't very robust. It holds up ok in the car but doesn't handle stress well. Stress means lack of maintenance, running hot in the mountains, etc. People that put bike carriers on the top of their Civics and go blasting down the highway at 75 mph are more likely to have transmission problems.
If you get a hitch and stick with your 350 lbs (which is about lightest trailer I've ever heard of), stay under 60 mph, and don't take the trailer into the mountains you might not hurt your car. What happens to many is once the hitch is on the car, the trailer gets either bigger, or gets overloaded and they end up towing much more than 350 lbs. Once you have the trailer you'll tend to take it longer distances, on interstates, and up and down some hills. That's when the car winds up dead on the side of the road.
If use your trailer you should have more cooling for the motor, more cooling for the transmission, and more robust CV joints, drive shafts, and brakes. The Civic needs upgrades of all these systems for and significant over the road towing.