From what I can remember, you can't save in GTA1. You get auto-saves by completing a city level. Each city has 2 levels.
Ok. Here's what happens. After you get a certain point total, a blue arrow comes up and you go to a guy and a cutscene plays, and that level is done. Each city is done twice, each time with different missions. So you have Liberty City: Gansta Bang, then below that you'd have the option to start Liberty City: Heist Almighty.
Then you would move to the next city. Repeat by getting the number of required points, and moving on and doing the city a second time. I think San Andreas comes next.
Then you do Vice City twice, and you're done.
Once you complete a city level, it will become unlocked in the main menu.