Three days are not enough for all of Rome, but they are enough for a coherent first encounter if you choose the areas carefully and leave time for travel.
Day 1: the historic center
Connect the central squares and monuments in one walking route. Avoid crossing the city for a single site.
Keep the evening for a walk without a rigid schedule.
Day 2: ancient Rome
Reserve a substantial period for the Colosseum, Forum, and Palatine. Check tickets and access conditions before traveling.
Day 3: the Vatican
The Vatican Museums require energy and selection. Do not add too many distant stops on the same day.
Practical note
Information about tickets, schedules, transport, and access can change. Check official sources close to your travel date.