Michael gave you some pretty common destination so I'm going to add some less touristy but awesome places around Milan.
Since I love lakes I would suggest you a visit around the many lakes you can find in the area, Lake Garda being probably the most famous one.
What to do
It is where the Italian writer Gabriele d'Annunzio lived after his
defenestration in 1922 until his death in 1938. The estate consists of
the residence of d'Annunzio called the Prioria (priory), an
amphitheatre, the light cruiser Puglia set into a hillside, a
boathouse containing the MAS vessel used by D'Annunzio in 1918 and a
Verona is a pretty nice and small city which was awarded the World Heritage Site. The city itself is beautiful and there's the famous amphitheatre and well.. the balcony of Juliet, of course, among many others thing you may want to visit.
One option you may consider is to stay somewhere in Verona or Desenzano and then do day trips to the places I listed (or others you have found yourself). You may also consider to ride a bike so you won't be bus-bounded (even tho the bus system in that area works pretty well).
Trenitalia is the official site for trains schedule. Milano - Verona is about 1h 20mins and it runs pretty frequently. A ticket is around 20 €.
The area around Lake Como is also very beautiful and closer.
Honestly, tho, If you have never been to Venice, I would spend the five days there since five days are perfect to really start to get to know the city, not just the most crowded touristy places. If you are lucky you can find hostels for about 20 € / night but you better book in advance. After visiting the most famous places just roam around the city and get lost, go to Castello, Dorsoduro, sestieri where people hang out the clothes outside (well, probably not with these weather :)) far from all the tourists. Venice is also pretty safe so roaming around night (less tourists as a bonus) it's highly recommended. Don't be afraid to get lost in those small calli, it's totally awesome. I love to just walk and look around and Venice is a good place to practice this degeneration but I don't know if it's the same for you. If you are more for a quick visit to the touristy places than you may want to stay in Venice two to three days and then move somewhere else.
If you are on a tight budget you may want to consider grab some food at the grocery store instead of going to the restaurant (eating something reasonably good for a reasonable price in Venice it's very difficult). Grocery stores that I know are:
- Billa (in Italian but pretty easy to figure out the address)
Local dishes you may want to try are:
Cotoletta alla milanese
Cotoletta alla milanese (milanese after its place of origin, Milan) is
a fried cutlet similar to Wiener schnitzel, but cooked "bone-in". It
is fried in clarified butter or olive oil and traditionally uses
exclusively milk-fed veal.
Risotto alla Milanese
Risotto alla milanese is made with beef stock, beef bone marrow, lard
(instead of butter) and cheese, flavored and colored with saffron.