Should I save over progress or always in a new slot when playing GTA V?

Depends on weather you want to replay a mission/start over from that point or not OR if you care about neatness.

If you save only on Slot 1 you will overwrite the previous save, you will not lose a single second of game play, you simply lose the option to start the game from that past point, instead you will start the game from the new save point. For instance you might want to save before the Assassinations missions, as sometimes the Stock market goes weird (I lost my investment actually) and if you DON'T overwrite your save, and save at Slot2 ,you can go back, with all your money and stuff and start over as if you never played.

On the other hand I save only on Slot 1 as I don't care about replay-ability (as I've finished the game before) and I used to have a problem on my GTA-SA following which is the save I want to use when I go back to the game after months of not playing.

TLDR: You generally save on Slot 2+ if you want to be able to replay a mission later, or you are more than one person using the Device, otherwise it doesn't matter.

the choice is yours to make but as you are just starting I will advise you to watch youtube vids about maximizing your cash otherwise you may well want to reload an old save. speciffically when you get to a point where lester appears and will give you assasination missions which you want to invest your money in the stock market with all charactures before and sometimes after such missions so as to get multiple millions in cash.

personally I find that as long as you do this multiple saves are not essential as most all missions can be replayed in the pause menu if for instance you are going for 100% with gold for every mission....