Travelling in Helsinki
For Taxi, use SMS, application or regular call for booking your driver. See details in http://www.taksihelsinki.fi/en/ordering-taxi . If you need taxi to pick you up from Tali DGP, tell the taxi center to send the driver to Talin huippu, to Talin frisbeegolfpuisto which starts from the end of Takkatie, next to Vermo horse track.
Most fluent and recommended is Uber. It works nicely in Finland, just departures from e.g. in airport can be tricky, but in those cases travel closer to Tali by public transportation.
Public transportation is very fluent in Helsinki are. Use Reittiopas http://www.reittiopas.fi/en/ and set Talin frisbeegolfpuisto, Helsinki as your departure or destination.
Also Google Maps works nicely.
Staying in Helsinki or Espoo
There are plenty of accommodation options close to Tali Disc Golf Park. The closest hotels can be found from Espoo Leppävaara that is within walking distance, and more options available in Helsinki city center. Look best options from Skyscanner, Hotels.com or use Airbnb. Eg. via Airbnb you can book a flat 69-98 euros per night (walking distance to Tali DGB), depending on amount of people staying.