How great an opportunity for you!! Italy is wonderful!! I have lived here for a total of 8+ years ( on and off) and it is a great place to visit. Any questions you have feel free to ask away. As for what you have already asked I will try to answer as best as I can.
Weather. Right now the temps in Venice are in the mid 40's; Florence the upper 40's low 50's; Rome and Sorrento in the mid 50's, but it is also a very rainy time. Come April you should find the weather warming a bit, especially in Rome and Sorrento. You can keep up to date on the weather through
www.weather.com - they are usually pretty close.
Clothing. I would dress for spring, but I would also use the mentality of "wrap and peel", meaning I would plan to dress in layers. What typically happens is the day will start off cool, then warm up quite nicely and then start to cool again in the afternoon and evening. Dress for comfort because you will probably do a lot of walking. Italians dress for everything, even to go to the grocery store. Very few italian woman wear sneakers, they tend to favor heels, sneakers are for the young, sports and some men. I would suggest if you have comfortable shoes to bring them, but if you need your sneakers go ahead most tourists wear them anyway. Sweatsuits, sweatshirts, sweatpants are not common unless someone is going to the gym or doing outdoor recreation, again I say the italians tend to dress most times. This was one of the hardest things to get used to for me because I love my comfort clothes. I usually wear a sweatsuit to go to the gym and even today when I come home from the gym and have to walk in my house I feel like all eye and fingers are directed at me. I would plan on casual clothing and bring something other than jeans for dinners in the evening, nothing fancy, just casual.
I have been to all the cities you have mentioned and I have enjoyed them all.
Sorrento is a beautiful seaside town and as someone mentioned the inlaid wood is beautiful. There is one shop in particular that caters to americans and gives good deals, but I can't remember the name. However, if you stand in the center of town by the statue with your back to the sea, walk to the right less than 1/4 of a mile, there will be a small alley way that takes you to a shop and the owners name is Jasmine, she is wonderful. If I can find the shop name I will let you know. If you have the time take a hydrofoil over to the Isle of Capri, you won't be sorry as it is absolutely gorgeous. As for food here I would be sure to try insalata caprese (fresh mozzarella (best in Italy) with tomatoes), calamari fritto (fried calamari) and after dinner a sorbeto (an icy after dinner apperitif made with local lemons).
Rome is one of my favorite cities in Italy and I will be going there the end of the month. As suggested above, take the tour of the cupola (dome) in the vatican, even though it is a hike up the view from the top is awesome. Word of advice, you can not take bags, backpacks etc. up so be sure you have enough file and batteries for your camera. You don't want to make the climb like I did and discover your extra film and batteries are in the backpack you left at the bottom with security. Rome has some wonderful sites, one of my favorites is Trevi Fountain, especially at night. There is a great gelato shop up from the fountain...fountain on your right, shop is up a little on the left...very yummy. When in Rome a typical roman dish to try is Bigoli Amaratriciana, this is hollow spaghetti in a sauce with tomato, onion and bacon and it is very yummy.
Florence is another great city, however, it isn't one of my favorites, but it is still nice and many others list it as one of their favorites. Florence is famous for their leather goods and for their florentine paper (stationery etc.) Word of caution if you are shopping for leather.....look the product over carefully and look inside and out to see that is is geniune leather (pelle/pelle genuine) and not "made in China". Don't be afraid to bargain, you may get a great deal. If you see a sign in the shop or at a market stall that says prezzo fissi, that means the prices are fixed and they do not bargain. Visiting the bell tower and going to the top is another way to get a great view of the city. Also in Florence you will find The
Disney Store, it is a nice store, but very, very expensive, especially with the euro to dollar rate being so poor. The not to miss food in Florence is beef depending on where you it is can melt in your mouth.
Venice is another favorite city of mine and I have been there many times, most recently last month for Carnevale. As said Venice is very pricey, I just read an article recently that said the locals are even complaining about the prices of meals out that they can't go out any more because prices are crazy. The most expensive areas are in Piazza San Marco and near the Rialto Bridge area. If you are able to get off the beaten path you can find better prices. For example, in San Marco square there is an ice cream shop that we went to, four adults and a child, we all had a sundae, a water and two coffee's and it cost us about $70. Another place in the square we got two coffees and it cost almost $15. Items of interest here are the venetian masks, some of them are very, very beautiful, Murano glass and lace products. Shop around because the a lot of the stores carry the same things and you can always find a better deal. There is also a Disney Store here, near the Rialto Bridge, but it is a lot smaller than the one in Florence. I can't recommend any Italian food here as we haven't found any we liked, however, there is an awesome chinese food restaurant near the Rialto that we love. I do have to disagree with one of the posters above in regards to a gondola ride, I think they are well worth it, especially if you are with a group of people. The cost is the same for one as it is for 6 and it gives you a different view of Venice and it is wonderful. We were a group of 5 and with tip we paid about $70 for a 35-40 minute ride. When you split that 5 ways it is very reasonable and what a wonderful memory you will have.
I hope this answers some of your questions. If you want more information or help please don't hesitate to ask. Also, if you want to see some pictures I have posted on the DIS with some of the places I have been just plug in tiggerlover and pictures in the search and you will see some of them.
Have a great trip!!!!
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) Really, weather and clothing have been so well-covered here you probably didn't need my reply to add to the mix-but I love Italy so much, I couldn't resist. I agree about the wear-and-peel layering for clothes (did that myself) and the reminder to have your shoulders and knees covered if you want to see the inside of the churches. (and trust me, you do-they are incredible) If I remember correctly, early April was beautiful while there seemed to be a lot of rain later in the month. But, I lived mostly in the Veneto regiong (Venice, Verona, Vicenza) and Tuscany (Florence) seemed a bit warmer and drier to me. 
. I hope you have a great time! The fodd as terrific, and I actually grew to like my water "con gas" (carbonated). We were there in early March and the weather was perfect, around 70 every day.
