I can't speak for sim racing games (as I never play them), but in arcade racers no two games behave the same way in terms of how manual gears work, in terms of drifting and cornering, so it's not really possible to give general advice that applies to all of them. So, for example, what works in Daytona USA won't work in Scud Race or Outrun 2 or Daytona USA 2 (unless, supposedly, you're using the Hornet in Power Edition but even then it feels different to me compared to D1...).
In terms of specific advice for Daytona USA, using individual buttons to shift is a valid way to play, and the ability to skip gears (i.e. 4th to 2nd) is something you can do with a proper shifter in the arcade and is how the game is meant to be played (and, as you asked, is something you can do with a real-life gear change as well). The idea in Daytona is to drop to a low gear and turn, to start initiating a drift, and the go back to a high gear when you've got the car at the correct angle to get the car to stop sliding and complete going through the corner without hitting a wall or coming off the track. The pros use a 4-2-4 (so going from fourth to second back to fourth) sequence with their gears to do this. I always seem to have a bit of trouble doing that though and find that a 4-1-3-4 sequence seems to work best for me. I'll post a video of me playing against a buddy to demonstrate it below (I'm the red P1 on the left). Using this technique you can do every sharp corner in both the Beginner and the Advanced course without having to touch the brakes once. The Expert course is another matter altogether though and a combination of brakes and gears is needed here I find (but TPH I've never fully mastered this course so others might disagree).
The drifting technique in Daytona USA 2 is a bit different and requires a combination of both brakes and gear changes. It's a lot easier to spin the car out if you're not careful.