Carb Heat -- On immediately
Then check Primer Locked.
If engine getting rougher for a bit after carb heat applied, ice is likely. It should smooth out once ice melted.
Mixture depends on high you are and temperature. The last time I flew a C-172 in the winter, there was quite the nasty stumble arriving overhead the airport after a descent at lower revs. Carb heat made it go away
Fuel pump can help or make things worse -- keep an eye on fuel pressure if playing with it. Even better, know what the normal fuel pressure in various flight regimes should be.
If still no cure, try each mag as mentioned before.
Oil temperature/pressure, EGT, CHT may all have hints.