Most budget trip's option is a bus. It is cheaper than train, and by time is going about the same. The main disadvantage is a necessity to spend 10-14 hours in sitting position. From advantages - bigger amount of stops in different Latvia's cities. Also buses got more destinations in comparison with a train. You can go to different cities in Europe by bus.
There are bus connections from Riga to virtually any point in Europe, Russia, Ukraine and Belarus.
Riga bus station (LV, RU, EN, GER)
Bus itineraries: quick search (LV, EN, RU)
Ecolines coach service (EN, RU, LV, LT, PL, DE, BG, EE)
Eurolines coach service (LV, EN, RU)
Nordeka coach service (LV, EN, RU)