There's no rail connection with Riga Airport yet, but plans have been developed to construct one. It's not yet known if it'll be a high-speed light rail connection or a 'normal' train connection.
Car and bus
Riga Airport is situated 10 kilometers west of the Latvian capital and is easily accessible by rental car. From Riga's center you cross one of the bridges that crosses the Western Dvina (Latvian: Daugava). All these roads lead to the A10 (E22) which passes the airport. The exit to the airport (P133) is clearly signposted.
Bus 22 connects the airport with Riga's city center. Minibuses, the Airport Express, also run to and from the Latvian capital. In addition, long-distance buses leave from the airport to neighboring countries, including Estonia, Russia and Poland.
Car rental Riga Airport
You can find counters of car rental companies in Arrivals Hall C. Five rental companies are present at Riga Airport: Auto5, Avis, Budget, Europcar and Sixt. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Taxis which all have a meter can be found outside the Arrivals Hall. The starting rate is 1.50 Lats (the Latvian currency which will be replaced by the € in 2012, € 1 = 0.70 Lats). Per kilometer you're charged 0.50 Lats. For a ride to Riga's center you pay approximately 10 Lats.
There are three parking lots near the terminal. P1 and P2 are for short-term parking; P3 is for long-term parking. From the latter shuttle buses run back and forth to the terminal.