Riga has only one airport International Riga Airport (RIX). From there you can get into the city by bus, which will bring you to the city centre, or by taxi.
You can also get to Riga by train or by bus. Information on schedule can be found on the following websites: www.ldz.lv and www.autoosta.lv.
Be careful not to use unauthorised taxi services – they might be ridiculously expensive. To take a taxi, either go to the taxi stand inside the airport or order by phone (some local numbers are +37167222222). The information on getting around the city can be found here.