been there a few times. Oficina do Duque is one of my favorite restaurants. There is also a bewpub around the corner (Duque Brewpub). The aquarium (Oceanario) is worth a visit, I believe its the biggest in Europe.
Check out Alfama, Pois Cafe is nice to get some lunch. If you go all the way up you should also visit the castle, nice view of the city. If you want to visit Belem, also stop for the LX Factory, its an old industrial place repurposed with some hip bars and shops and a pretty cool bookstore.
Take a day to visit Sintra, the Moor Castle is impressive. You can buy combitickets to check multiple things there. Also, let yourself be driven up for 5 euros each, way to hot to walk all the way.
You can also grab a train to Cascais and hop out at one of the beaches where the train stops.
I also like to just buy some drinks and snacks and chill out at one of the viewpoints like Miradouro de Santa Catarina, usually there will be a band playing and it's nice to just chill out a bit and admire the view.
Also, it will be very hot, so make sure to drink enough, and there is a lot of elevation in the city so just take it slow.
edit: seems to weather is much more pleasant than in mainland Europe this week, you should be good :)