How to make fluffy rice and which type of type of rice to use?
As is mentioned in this question (and this one), one of the keys to keeping your rice from clumping is to rinse the uncooked rice with cold water until the water runs clear. This rinses off the starch on the surface of the rice. The starch can cause your rice to stick together. 

Some types of rice are fluffier than others. The longer grain rices tend to have lower starch (and, of course, Arborio rice has so much starch that it makes for the characteristic texture of risotto). So even if it's not Jasmine or Basmati, they're probably using at least a medium grain, rather than short. 

A quick Google search for Greek rice yields a lot of recipes that suggest sauteing the rice in oil before boiling it. This is a common technique for pilaf, and adds a nutty sort of taste. I'm not sure it'd have any effect on stickiness, but it might help with flavor. And you can also saute onions and other vegetables to add flavor.