You are right, Uber is not operating in Denmark (long story short: they didn't comply with the Taxi law). We do have plenty of regulated taxis though, so if you want a taxi, that's no problem.
There's a lot of sights in Indre By (Inner City), but it's also very expensive. You can possibly save money by looking at other areas. Public transport is good and efficient, so no matter where you stay you can get to the city centre.
Here's a very(!) general characteristic of the different areas in Copenhagen:
Indre By: central area (with places to shop, eat and drink), expensive, many sights, historic old buildings.
Vesterbro: Small RLD area (around Istedgade), hipster neighbourhood, cool restaurants, bars.
Frederiksberg & Østerbro: Mid-to-Upper class residential areas, nice cafés, nice parks.
Nørrebro: International vibe, more ethnic restaurants, sort of a hipster feel.
Nordhavn & Ørestad: Newly developed modern residential areas (not much to see here as a tourist IMO).
Amagerbro: More mixed, not a big tourist area.
As long as you stay close to a train or metro station, it will be easy to get to/from the airport. www.journeyplanner.dk is your friend when travelling on public transport, it makes all timetables searchable and show you some ticket prices. More info about tickets can be found on www.publictransport.dk
For the record, there are no unsafe areas in Copenhagen - so you don't need to worry about that when booking your hotel.