A good bus service connects Helsinki airport with the city centre, and is well supplemented by good value taxis.
The best way of getting around the city centre is by buying a Helsinki Card, which will allow free travel on all buses and trains in the centre, and also entry to the many museums and galleries as well as a complimentary city tour.
Throughout the city, you can pick up free citybikes to get around on (you pay a deposit) and cycle lanes run around most of the main attractions.
Hiring a car in Helsinki is a good idea only if you plan to leave the city, because parking can be a problem - although you'll rarely see a traffic jam because most of the locals stick to public transport.