How do I find the registration or serial number for the aircraft I flew on specific trips in the past?
The US (if that's your area of flying) is very kind in publicly releasing stats compared to other territories, but the answer is still you can't without a paid service.

See:


Where can I get information on ATC facility activity, like the number of flights handled in a month?
How can I get historic flight data by N-number?


In short: The FAA only keeps broad records that do not contain information such as tail numbers.

Start a personal flight journal for future flights, it's a hobby some enjoy. I did at one time, and when I couldn't catch the registration, I asked the flight crew.