Milan is not too far from Pavia, you can easily spend 2-3 days for just the city sights. There is a Duomo in every city but the one in Milan is a must see. Castello Sforzesco, the Church of Santa Maria delle Grazie (Here you will find the Last Supper painting made famous by the movie Davinci Code, make sure you buy the tour/book you entry in advance), Piazza della Scala (Opera & Ballet performances) and Galleria Vittorio Emanuele (fashion district is closeby) are all worth a visit.
Lake Cuomo/Lake Maggiore area is great weekend getaway from Milan, here you laze around/cruise around some great lakes and soak in terrific Views of the Swiss Alps. (You can get a glimpse of the 'Lifestyles of the Rich & Famous'). Venture further north into Lake Lugano in Switzerland if you have time.
Vegetarian options are a natural component of Italian cusine. Luckily most places even very small towns you will find trattoria’s, or pizzeria’s where you can always get slice of plain pizza with eggplant or tomato. The more formal and more expensive dining options are called Ristorante and least expensive the pizzeria. Antipasta/ antepasta are appetizers served before the main course and they always have some vegetarian options. You can always ask for plain pasta with Arabatia sauce (to get some spice in there).
If all else fails there are some of the best gelato’s sold in gelataria's found in every streent corner or Piazza (place people hangout).
Like most places in the west Vegan could mean you eat fish and avoid dairy/other meat products. Something to be careful about.
If you have not finalized your accommodation options, check out http://www.venere.com for some great homestay, B&B and long stay apartment options. (Given that your stay is for a long 4 weeks you can rent an apartment with a fully equipped kitchen). Cooking your own meal in the evening is a great vegetarian solution.
have fun..., italy is a place you will want to keep going back...
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