First of all: don't forget that shops are closed on Sundays!
All trains from Brussels and Ghent also stop in Berchem.
I wouldn't bother with trams in Antwerp. Just get a daypass for Velo. You can find Velostands at nearly every corner and they are very convenient. You can buy the pass at any velostand, just read the instructions. Antwerp is very much a cycling city.
If your sons like architecture than you definitely should walk through the Cogels-Osylei, the most beautiful street of Antwerp and right next to Berchem station.
Zurenborg: go to the Draakplaats or the Dageraadplaats. There are many bars and restaurants. My favourite ones are Bellini for pizza, Picci for pasta, Humm for vegan breakfasts and lunches, Byzanaat for Arabian food, Het Strand van Oostende for anything from burgers to classic Flemish foods, café 't Zeezicht, a very popular pub.
I live in Borgerhout, a very multicultural neighborhood, that's getting more and more popular. The area is being nicknamed Boho and has a more artistic vibe. Places to be are: Clo (vegan lunches) in the Kroonstraat (between Zurenborg and the centre of Borgerhout), Bar Lucca (small sharing plates) and Mombasa (a bar dedicated to cycling) at the Moorkensplein, Bakeliet (vintage bar with live music) at Laar, and Frituur Tina (voted to be the best frietkot in Flanders this year), Bar Leon and Borgo Gelato. These last 3 are situated at the Krugerplein.
Another nice place is Bar Chapel , a pop-up bar in the garden of an old monastery in Kerkstraat.
All of these places are not in the centre of the city, but they are very popular.
For jazz: café De Muze in de Melkmarkt and Den Hopsack in the Grote Pieter Potstraat. These are in the centre of the city. If you go to the centre you have to visit the Carolus Borromeuschurch. It's the most beautiful church in Antwerp and it's on the most beautiful square in Antwerp. Right around the corner is the Wolstraat with antiques shops. Also nearby is restaurant Barboeuf, which has typical Flemish food and really nice steaks (even horsesteaks if you dare, they are actually really really nice, but I guess most English people are abhorred by the idea)
Another must-see in the city centre is the UNESCO World Heritage Site Platijn-Moretusmuseum. It's an intact printshop owned by one of the first European printers. It's absolutely stunning! One of my favourite places to visit.
Another nice area is the Zuid, with is a bit more expensive. There are a lot of fashionshops and the new KMSKA-artmuseum, which is really worth a visit. They also have a brand new park (it opened last weekend). Lots of bars and restaurants along the Waalse and Vlaamse kaai.
And maybe your sons might be interested in the voetgangerstunnel to Linkeroever. It has the oldest escalators in the world. They are art deco, made from wood and still in use. The are situated between the city centre and the Zuid.
A last neighbourhood I can recommend is the Eilandje. If you want to walk along the water, this is the place to be! You can eat along the water, visit the MAS-museum of the Red Star Linemuseum, which is dedicated to all the European emigrants who left to the US or Canada through Antwerp.
Parks: stadspark will be the nearest to you. It's also rather small.
Middelheimpark is an open air museum for sculptures. It's free!
Rivierenhof is the largest park. Go and get a drink in Bar Stark and sit in their garden.
Another popular park is Park Spoor Noord where you can take the iconic cycling/walking bridge to the Eilandje. There is also a big bar in the middle of the park where you can eat.
As you can see, there is enough to visit and do in Antwerp alone to fill 3 days. You won't have time to go to Brussels or Ghent 