You are in for a treat as there are lots of places to explore. It’s a great walking city but lots of hills so your cardio will improve! You can also take the historic trams to get around, although the metro is probably faster and with fewer tourists. There is even a cable car linking the aquarium to the redeveloped eastern dockland area. Ubers are plentiful and cheap! I recommend getting a rechargeable transit pass which covers most public transport. Many newsagents sell them.
Places to stay
I would suggest heading to either the Chiado or Principe Real. They are quite close to each other, so either will do – centrally located and you can get to the other areas easily. They felt very safe (as is Portugal in general) and have a great vibe with lots of interesting shops, restaurants, etc.
A favourite hotel so far which is close by is Britania Art Deco – a restored boutique Art Deco hotel with lovely rooms, excellent service and a great breakfast. A bit pricey but sometimes they have deals so check it out! Another boutique hotel to recommend is in the Rossario district is Casa Ruy Lopez. It has a small pool and is organised around a walled courtyard – a bit out of the main core but not too far. The hotel was quite cool, to the west of the center of Lisbon and surrounded by a lot of embassies. You can still get to the central part of the city on foot.