The main set of mountains for hiking when starting from the city centre are the 2/3/4 (depending on how you count!) to the north-east of the city centre, including Fløyen (the one with the funicular railway) and Ulriken (with the cable car). There are a decent number of proper walking trails up and around these.
The trails are largely divided into "green" ones, which are well made paths and quite flat (I believe which often get used for cross country skiing in the winter), and dotted black ones, which are often quite steep and not made up.
Two things to be aware of navigation wise - there aren't that many map boards posted, and while there are proper signs at each junction they are by name but without distances. Unlike similar paths in America, there aren't distance markers as you go, and unlike similar ones in the UK there aren't distances at the junctions. Map boards were a lot less common than in similar areas in other countries I've hiked in.
If you just want to hike up Fløyen, I'd say you're fine with just a local tourist map - they normally go to about there and have most of the walking routes. For a bit more than that, you might be fine with using a local tourist map to get you to the start of a trail, then take a photo of one of the big board maps you find at the start. For anything else, get yourself a map before setting off. There are some great hikes very close to the city, some lovely lakes for swimming in, stunning viewpoints etc, and it'd be a great shame to miss out on any of them through not having a map to find them!