Aside from the parks and bars there's loads to do and see in PoA. I'll start with the city centre. The Public Market, Mercado Público, is well worth a visit. The 19th century building encloses a courtyard, recently roofed, containing several stalls which sell everything from herbs and spices, fish and meat, cheeses and wines to local offerings for umbanda ceremonies. The upstairs area has bars and restaurants, good for a choppe which is cold draught lager. Unfortunately, like all the city centre area, it is dangerous at night and therefore closes early (9pm) and no one hangs around. Even during the day you have to be careful around the city centre, don't leave your bag or camera on a table unattended, don't even turn around for a moment, keep your stuff close. There are street urchins always on the lookout for careless people, and they're very fast on their feet!
Just across the square from the Public Market is the Chalet Bar, there is a good open air patio and an upstairs terrace and they serve good food and choppe.
Rua dos Andradas, popularly known as Rua da Praia (yes, doesn't everything seem to have two names, an official name and a popular name!), is the main shopping street, crowded and hot, right in the centre there's Praça da Alfandega, where the annual book fair is held and further down there is Casa de Cultura Mario Quintana. The remodelled Hotel Majestic, CCMQ has lots of art and cultural events, with a great café on the top floor and an equally good one on the ground floor in the hotel arches alleyway between two streets.
From the docks right in the city centre a boat trip on the Cisne Branco is great, it takes about an hour and goes through the channels between the estuary islands, costs R$10 (Brazilian Reals) per person, that's about 2 pounds sterling. The contrast between the rich and the poor here is incredible to see. Look on one side and you'll see an beautiful mansion with extensive lawns stretching down to the river where there is usually a jet ski and launch tied up at a dock. Look over the other side, just 20 yards or so, and there's a delapitated wooden shack, half a dozen brown naked kids hanging around and piles of garbage where the people have collected and dumped to sort for anything recyclable (they get a meagre earning with that).
A 15 min walk from the city centre will bring you to the Gasômetro, the old power station, now restored and used for cultural events. All along the dock area from the centre to the Gasômetro are the disused quays, there is a plan to convert it all to culture area, but that plan has been on hold for years. Another boat trip goes from the Gasômetro, a little cheaper than the Cisne Branco. There is also a "floating bar" here, a kind of large two-decked platform with a bar but unfortunately it's not very well looked after and of very poor quality. It's not bad for drinking a cold beer and watching the sun-set though.