Personal recommendations for friends who were visiting me in Navigli. All the links take you to Google Maps – I recommend saving places and then clicking “directions” for travel options.
Obvious touristy things

The Duomo (above) and the surrounding area. It’s right next to the very pretty Galleria Vittorio Emanuele II, and 2 famous department stores – La Rinascente (The Renaissance
– not as good as Harvey Nichols, in my opinion, but with cool top-floor cafés), and Excelsior (smaller but more interesting).
Otherwise you can walk from Piazza Gae Aulenti (the heart of Milan’s recent development), down Corso Como – stopping to have a look inside the famous No. 10 (a very fancy café / clothes shop / gallery combo) – and finish up at Eataly in Piazza XXV Aprile (an enormous food store with a gimmiky name, but that’s actually popular with Italians – it selects the best locally-produced food from around the country).
Galleries

Pirelli HangarBicocca (above)
Aircraft hangar converted into a free contemporary art space. Open until 10 p.m. Thu-Sat. Not remotely near anything else but worth the trip.

The Triennale (above)
One of my favourite places in Milan. You have to pay to access most of the exhibitions but usually a couple are free.
Fancy new exhibition space with newly opened tower (I prefer Hangar). Open until 9 p.m. Fri-Sun. It has a bar designed by Wes Anderson, which is not as cool as it sounds: he wanted it to look like an perfect replica of a traditional Milanese bar, and did such a good job that when I first went there I thought they’d simply built the Foundation around the bar. If you want to see a traditional Milanese bar there are plenty of real, prettier ones in the rest of the city.
Apparently all the major fashion labels have exhibition spaces. Walking distance from the flat. Haven’t tried it yet.
“Culture museum” (“MUseo DElla Cultura” => MUDEC). Cool building but you have to pay to see most of the exhibitions.

Spazio Rossana Orlandi (above)
Incredible hidden space for contemporary design (every item of furniture costs at least €20,000). I don’t know if you can get it as an ordinary member of the public – I’ve always gone with designers.
Apparently Michelangelo’s The Last Supper is somewhere.
Beautiful places to walk

Navigli (next to the flat – above). Don’t walk too far south / east, away from the Darsena, where the canals meet – an area a Milanese friend described as the Bronx
.
Or a bit further north into Parco Giovanni Paulo II.
Brera is the picturesque centre of the city. It’s in between Eataly and the Duomo, and near Via Montenapoleone, so you could tie all those things together.

Parco Sempione (above) is really pretty – behind the castle and near the Triennale.
The fashion district
Walk down Via Monte Napoleone – way fancier than Bond Street.
Restaurants

Trattoria della Gloria is walking distance from the flat, and is a great, traditional restaurant. Apparently Coke Pizzera, just the other side of the canal, has the best pizza in the area (I haven’t tried it yet). In the centre, La Drogheria is very good, and has a cool downstairs bar (you’ll need to book). I’ve been recommended Osteria Delbinari (above) but haven’t been yet (it looks expensive). If you’d like reasonably-priced local fast food, Flower Burger is a great vegan place – there are other branches throughout the city.
Bars

Nearby, Morgante Cocktail & Soul is very pretty (maybe too pretty). Mag Café is supposed to have some of the best cocktails in the city. Rita is famous for aperitivo (pre-dinner drink + snacks). In the centre, N’Ombra de Vin (above) is cool but expensive.
Clubs that don’t suck

Rocket (above) is walking distance from the flat. Young, friendly crowd. Friday pop, Saturday hip hop. Alternatively there’s Old Fashion, next to the Triennale, which has a large outdoor section.
Practicalities
Carrefour is the local supermarket, open until midnight. Essenlunga is larger but further away. Bio c’ Bon is the local organic / health food shop. If you’re catsitting you might need Number One for Pets (nearer) or Animalia (larger).

If you need a café to work in I strongly recommend Gogol & Company (above) – 20 mins walk from the flat, with very friendly staff with good taste in music. Alternatively there’s Moleskine (great coffee but the WiFi never works) or Libreria Verso (nearer – I’ve never tried it but it looks nice). Or if you go to Coffice you pay by the hour (€4), with coffee and snacks included.