Riga International is the largest airport in the Baltics and is owned by the Latvian government, they have 3 terminals, one for incoming Schengen flights, one for outgoing Schengen flights and one for None Schengen flights.
I had the pleasure of flying to Riga International Airport from Glasgow Prestwick last year, after a rather uneventful flight we arrived in Riga International smoothly at around 10 pm. The walk from the tunnel leading from the aeroplane to the passport control was quite long for a small airport but the passports were checked quickly and our bags were already waiting for us.
I was very impressed with the efficient service, I then headed outside where a friend of mine was waiting to meet me. We took bus 22 to the centre for about 40 santimu (40p), there was an Italian guy who had got the bus without any Latvian money and the butch women selling the tickets wasn't particularly pleased about this but eventually let him ride anyway. They leave every 10-30 minutes and it takes about 30 minutes to make the 13km trip to the centre. You may have to pay more for very large luggage about 1 lat 20 santimu (1.20 pound*). If you wanted to take a taxi instead, expect to pay about 8 lats.
I didn't notice a particularly good choice of shops but there are some, however I wouldn't get your hopes up!