Do NOT stay near the railway station - bit of a dodgy area.
The Hotel Raphael, just behind Piazza Navona is lovely.
Or, if that's too expensive, try Albergo del Senato, opposite The Pantheon. It's a bit old-fashioned, but comfortable and a fab location. Ask for a room overlooking the square.
Also, be aware that hotel rooms in Rome are are generally quite small, so book the biggest you can afford.
Staying near the Vatican is a bit too far out for all the non-Vatican things.
Trastevere is nice, and a not so grand as some areas.
Stay centrally, and you can walk everywhere, but take VERY comfortable shoes.
Book major sites like the Colosseum at least 4 to 6 weeks in advance.
Don't pack your days too full, as you want time just to sit in cafes, wander down back streets etc. Four days is just a start - you absolutely can't see everything, so be selective. Plan in advance, keep planning and plan some more!