Super Mario Bros is a good choice. The game is not too long, not complicated and all the tricks are already discovered.
You won't beat the current record, but you will learn the basics of TASing.
I would suggest you to:
1) learn an emulator TAS tools first: savestates, frame advance, movie recording + maybe TAS Editor (TAStudio in Bizhawk) and RAM search, but these are for advanced TASers.
2) watch some videos about how to make a TAS. Here some examples:
- adelikat's tutorial series (terrible video quality):
Part 1:
https://www.youtube.com/watch?v=OS75JLwJExk
Part 2:
https://www.youtube.com/watch?v=j7EozJIIWME
Part 3: ???
Part 4:
https://www.youtube.com/watch?v=x9MulJjsk-U
- dwangoAC's tutorial:
https://www.youtube.com/watch?v=6tJniMaR2Ps
3) make a TAS of Super Mario Bros using your knowledge and learnt material from videos;
4) compare your TAS to
HappyLee's TAS and see which tricks your have missed;
5) try to figure out how they work (check submission texts of some movies mentioned in "History of this entry" section and the
forum thread) and represent them in your TAS #2.
It would be great if there was a video of someone explaining how to find various glitches with all the theory included (not explaining how they work, but how to FIND them), but there isn't any (that I would find decent), so you should mainly count on your analysing skills and other people who has courtesy to answer your questions.