Milan in early April?

I've not been to Milan but have some places on my bucket list.

Santa Maria delle Grazie(Last Supper)
Duomo di Milano
Brera District
Chiesa di San Maurizio al Monastero Maggiore
I Navigli

Also, If you are a clothes horse... Milan is one of the fashion capitals of the world:)

DaVinci's Last Supper is of course a world treasure, even if you only get fifteen minutes inside the room with it. Note tickets sell out quickly and get eaten up by third party tour operators. Nevertheless, local tours that include these and other places in the center of the city are available as a package. We saw the Last Supper, the Duomo, La Scala, the Galleria, and the Sforza castle on such a tour

I am NOT a shopper but Milan shops were hard to resist, especially shoes.

I will be there in mid April and I also hope to visit the Lakes area

We visited there April 2016. Of course the Duomo and Last Supper (make your reservations asap!). I think the Last Supper is the most amazing piece of art I've ever seen.
We didn't linger in Milan, though. You can see it quickly. Then we took a train to Lake Cuomo. We stayed in Bellaggio-the lakes are breathtaking! You can go up even for one night, and get a taste of the lakes. I would do 1 night in Milan and one night in Bellagio, if you are up for a little adventure. And if you have 2 nights. Not sure how hectic the rest of your trip is.

N Milan is much, much more than the Duomo, the galleria and the Last Supper. It has many interesting and beautiful places to visit.
You can start at the Duomo, walk through Brera, One of the first neighborhoods of Milan, now filled with unique shops, galleries and restaurants, arrive to the trendy Corso Como and then unto Porta Nuova, the new post of the city, where you can visit Piazza Gae Aulenti, and see the innovative architecture in the area...
You can go to the Navigli, antique water canals surrounded by restaurants, bars and shops. Stop there for aperitivo and dinner.
When you go to the Last Supper, walk along Corso Magenta, and you'll find traditional "botteghe", old shops, bakeries, barbershops...

There are many options, it's just that people don't venture much away from the Duomo.

I really like Milano and have been there many times. Milano is not one of the more popular destinations in Italy, but it is one of mine. Here is a list in no certain order:
1. La Cenacola-"The Last Supper"
2. Duomo and its roof.
3. Navigli section for nightlife.
4. Galleria Vittorio Emanuele
5. "Il Toro" inside the Galleria Vittorio Emanuele and spin your heel on the bull's balls for good luck. The Milanese have a sense of
humor. Make sure you take a picture.
6. When in the Galleria from the Piazza del Duomo, walk to the opposite entrance, turn right, and walk down the street until you see
Luini's on the left for a panzarotti. A panzerotti is fried dough stuffed with goodies like ricotta and spinach, etc. it's easy to find
because because it's so crowded.
7. Peck's Gastronomia and it's upstairs cafe.
8. The Leonardo Di Vinci Museum and make a reservation to tour the Italian Navy submarine.
9. The Osteria Mama Rosa in Piazza Cincinnato for the best clam sauce on the planet.

Buon viaggio,

If you like this sort of thing, check out the Monumental Cemetery (Cimitero Monumentale) for the sometimes hilarious, sometimes weird, always interesting, melodramatic, idiosyncratic, life-size sculptures on the graves. Two full-size oxen pulling a plow with much difficulty down a slope and clearly bellowing about it, for instance. We spent 1/2 a day there and didn't see it all. It's free and park-like and peaceful.

I was in Milan the end of March a couple years ago. The city park - just behind the castle was beautiful at that time of year. I have been to Milan previously so that trip I did a lot of day trips but you might get some ideas from my blog post (which has lots of photos). Most of the Milan stuff is at the end of the post.
https://andiamo.zenfolio.com/blog/2017/5/march-in-northern-italy-milan-genoa-lake-como-lake-lugano-pavia