In my experience, Turn Signal problems are usually one of these;
1) Bad bulbs (never hurts to check them "just in case")
2) Hazard Flasher
3) Electrical component of the ignition switch.
4) Turn Signal switch (usually the cancel function goes but I'd check the connector and the switch as a last resort)
Since you just bought the car you really need to start from the fuses (just me but I always start with fuses) to the bulbs, to the the hazard switch to the ignition switch. All the work is simple, but parts are expensive if you just replace good ones with new ones.
Despite the fact that your hazards are working it is so common that I wouldn't automatically rule it out. I've heard wiggleing the key with the turn signal on my get it working (obviously indication a shot switch) but when mine when it was immediate and total. Can't hurt to try though.
How many miles does it have?
Last edited by donf; 10-16-2014 at 05:56 PM.