As for transportation, there are frequent trains between the two cities. Check on the website of the Belgian Railways what discounted tickets are available. Usually, on weekends these tickets can be used after 9AM, and are limited to local trains only. But the distance between Brussels and Antwerp is not so great that it will make a major difference in travel time. Check in your hotel which of the various Brussels stations that handle the Antwerp trains is the closest.

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The train from Brussels to Antwerp takes between 35 and 45 mins. There are trains every 10/15 mins. A return ticket costs 13.80 euro on weekdays, 7.80 euro on weekends. A senior return ticket (65+) costs 5.30 euro (you can't travel before 9 am).

From Antwerp Central Station you either walk or take the metro straight to the historical centre. It's a 20 min. walk along the main shopping street (de Keyserlei and Meir). Don't forget to look up to the beautiful façades! The historical centre is small, so you can see a lot in a day, unless of course you want to visit all the museums. A 'must see' IMO is Our Lady's Cathedral, the Grote Markt (Market Square) and the surrounding streets, and maybe Rubenshuis Museum or Rockoxhuis Museum. The latter are nice examples of how the rich used to live in the 17th century.
Furthermore sample a few beers in one of the numerous cafés on the squares.

The Antwerp Zoo is a pretty zoo, a heaven of piece in the city. It is next to Central Station. I'm not sure, however, whether a short visit is worth the entrance price (19.50 euro per person, 15.50 for 60+).

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We took a day trip to Antwerp from Brussels last fall. See Day 8 of my trip report. I would have liked a second day there to see other churches and museums we missed. We didn't see the zoo and it would be pretty low on my list of attractions.
Plan to spend some time in the train station admiring it--it really is beautiful.