My favorite park is the one around via Palestro, not too distant walking from your hotel. The gardens around nearby Villa Reale are also a possible visit. The Giardino della Guastalla is an different and somewhat secluded area, also another possibility.
In spite of having lived 12 years in Milan and having a Milanese wife, I find difficult answering your other questions. The Milanese soul is elusive and somewhat different from the general Italian soul. The first things you will notice about Milanese is that they are in a state of perpetual haste to go somewhere, and that good and cheap restaurants are more difficult to find.
If you want some unconventional destinations:
Chiesa di S. Bernardino alle Ossa - via Larga - a chapel of the church is entirely decorated with bones and skulls from the cemetery of the former ancient hospital that was very close and now is the University building. A gothic nightmare.
Tram ristorante ATMosfera. Have a dinner while roaming the city on a 1928 tram (the model is very similar to some San Francisco trams, as the builder was the same)
Visit the Giuseppe Verdi tomb housed in Casa di Riposo dei musicisti (an home for old retired musicians)
