First driving lesson

Today was the very first time for me to drive a car. I passed a driver's written driving test last week so that I could start practicing how to drive a car(!). Since I lived in Japan which has good public transportation systems, I never thought that I would try to get a driver's license. But I got started to feel that it would be necessary to get a driver's license after I moved to America.

"Necessity" is a noun. If you want to use it, it is better to say: "I started to feel that a driver's license would be a necessity after I moved to America."
 
Driving a car is such a huge responsibility!
I was exhausted even though I practiced in a parking lot in the university which was totally vacant due to the national holiday.
I hope I can make a progress in driving a car and pass the skill test within a month!

Good luck!! I need to get my license soon too. Driving a car is a little scary for me. You're right, it's a huge responsibility!