In order to save the game in Grand Theft Auto Vice City you need to go to a save game point. You cannot save the game via the pause menu.
Save game points are located around the city and can be found whenever you buy a property. If you have just started then your first game save point will be at the ocean view hotel. This is the hotel near the start if the game.
One if the early missions in the game will show you how and where to save the game.
Game save points is marked on the map with a pink and black cassette tape icon.