There is no train connection to the airport, but a bus goes to the train station in Innsbruck. From this station there are trains that go to places such as Seefeld, St. Anton and Kitzbühel.
Car and bus
The airport is directly along highway A12 that goes south of Innsbruck. The exit to the airport is clearly indicated (Innsbruck West/Flughafen). From Germany (border crossing Mittenwald/Scharnitz) you do not have to pay a toll if you are going to the airport, then you take the exit to the Bundesstrasse 171 (Völs Kranebitten/Flughafen).
City bus F goes every 15 minutes to Innsbruck Train Station, tickets cost €1.50.
Car rental - Innsbruck Airport
On the ground floor of parking garage 3, you can find counters for car rental companies: Denzeldrive (National/Alamo), Hertz, Avis, Budget, Sixt and Europcar. You can also reserve a rental car in advance using the service EasyTerra provides, they compare the rates of several providers in a single quote.
Taxis are in front of the Arrivals Hall which can take you to the center of Innsbruck in 10-15 minutes and costs approximately €12.00. The airport website has an overview of taxi prices to other destinations.
There are two parking areas (P1 and P2) and a parking garage. The first 30 minutes are free, and after that you pay €1.30 per hour with a maximum of €13.00 for the first day. A week of parking costs €57.00 and two weeks costs €90.00.