great question! For only a few days, you will want to hit the usual: L'Acadamia, The Duomo/Baptistry and the museum that is around the corner (has yet another one of Michelangelo's pietas), San Lorenzo/Capelle Medici and Santa Croce and Santa Maria Novella. You can make a very nice and efficient stroll from ~San Lorenzo towards the Duomo, then off towards the Old Square via the Old Market (The new, "dirty" market surrounds San Lorenzo, so you can experience that, too), cross Ponte Vecchio to the other side of the city. You have to do a bit of a stroll to make it to Santa Croce, but it's worth it imo (lots of famous dead people and their tombs).
You should be fine in April, so you could probably get into the Uffizzi without wasting an entire day standing in line. If you can, do it. It's small enough so if you can get through without waiting, it would only take a piece of a morning or afternoon. Check on the schedule there before you arrive or first off and plan around that if you can.
Don't eat off of the main streets. Walk a few blocks off and find some Enotecas or Lunch bars where they sit you down at a shared table with a bunch of Italians and ignore you most of the time. The food is generally better on the Alta Arno (Santo Spirito area, I think)side of the city...because it is less populated by tourists, I guess.
For the classic view of the city, you want to take a bus out and up to San Miniato Al Monte. the large patio outside gives you a gorgeous view of the whole area. Leonardo actually built some cannons and placed them up there at one point to defend the city from the Duke of Milan, or maybe it was the pope and one of his armies. I forget.
But I'm all into that type of shit, so Florence is perfect for me. It's like 10 times what Rome is but less than 1/4 the size. Food is better either way. Plus there is Sienna and all the other towns relatively close, but you probably don't have time to venture out there if only a few days.