If you're looking for a green area to spend some time in the sun, go to Park Spoor Noord. A good place to play with the kids, go for a pic-nic or just laying on the grass.
Every Sunday, in the city center, you have the Vogelmarkt, which is a market a little bit about everything: clothes, food, animals. Since most shops close on Sundays, this is a nice thing to see, when strolling in the center.
The most known street for shopping is the Meir, where you have the normal brands (like Zara, H&M, C&A, etc) and then some that are not so common in continental Europe (like Forever 21, Urban Outfitters, River Island). They close on Sundays, but on any other day the street is pretty crowded.
On that same street, you will find The Chocolate Line, which is one of the most famous chocolate stores in Belgium. Even if you're not planning on buying anything, just enter to see their delights and to enjoy the space where it is located, as it is a very nice and fancy palace.
The Central Station is very beautiful too, both on the inside and on the outside. If you go through the back entrance/exit of the station (Lange Kievitstraat), you will pass by lots of the famous diamond stores and will be in the center of the Jewish neighbourhood. You can see lots of orthodox Jews and their families passing by, all dressed up traditionally.
Not as famous as Amsterdam's red light district, Villa Tinto is, on the other hand, known to be the world's most hi-tech brothel. It is located near the pier and basically is a shopping mall turned into windows for prostitutes. It has a police station on the inside, lots of security and modern technological features. It can be interesting to pass by there (you can cross on the middle of it and the adjacent streets are also from Antwerp's red light district) just to see this one-of-a-kind thing.