Go to Bologna! Eat at Quindici, Osteria Broccaindosso and have gelato at Sorbetteria Castiglione.
Except for the Uffizi, Florence is overrated.
Tell me more. Where should we stay, fly into, etc. Really I'm floundering over here.
Usually Rome and Milan are the cheapest to fly into because there are the most flights and options. But still check aroind - try Florence, Bologna, Venice, Pisa, Naples even. There is SO much to see in Italy that you just can't do it all on one trip, so my advice is to pick a region. I am partial to the North, personally, but you can't go wrong with pretty much anywhere (well Puglia probably isn't the highlight of the country). If you are reasonably brave, I'd recommend renting a car and driving through Tuscany. April is the perfect time for this - it's gorgeous and the weather is lovely. See Siena and Pisa for sure (Pisa has the tower, yeah, but it's also a lovely college town, underrated IMO). If you haven't been there, you kind of have to see Florence - see the Uffizi, see the Duomo, see the Accademia, see the Ponte Vecchio. But my personal opinion is that the city is uncomfortably touristy.
You could also do the eastern side - see Venice, Padua (totally overlooked by most tourists but it's beautiful and its directly on the train line from Venice to Bologna and onto Florence and Rome). Maybe take some side trips to Ravenna (famous for some stunning Byzantine mosaics some of them hauntingly submerged underwater) and Faenza which is just a cute and beautiful little town (not necessarily a must-see but still a nice place to stop in and have lunch if you're nearby). And of course Bologna - it's famous for its food so that's the place to eat. There's also a huge (and very old) university so lots of cheap spots to eat too.
Or do Rome, which could be a whole trip in itself, then take a train down south to Sorrento and Capri and the Amalfi Coast and see Pompeii and Ercolano. April will be a beautiful time there too.
This summer I did Lake Como and the Cinque Terre. Lake Como is the most romantic place I've ever been. Loved it so much. I would fly into Milan, go to Lake Como and then up to Switzerland if I did it again. Although, the Mediterranean was also lovely (I didn't love the Cinque Terre as much as some of the nearby areas).
Previously, I did Venice (loved, but it was off season), Bologna (didn't care for), Florence (good but didn't love it as much as others do), & Rome (loved). If I did that trip again, I would cut out Bologna and spend the majority of my time in Venice and Rome.
Italy is freaking amazing. You really cannot go wrong. And you are going to have to go back.