In my experience as an American:
We do use the word delicious fairly often. It is not at all unusual or strange.
Its use compared to the more general words "good" and "great" is dependent on context.
For example, if my friend asked me "how's the food?" and I enjoyed the food, I'd more likely use "it's good!" or "it's great!" and reserve "it's delicious!" only for when it's exceptionally good. And this is mostly the case, that "delicious" is simply "more good" or "very good." A similar word would be "scrumptious." It's used by many, but it's just less usual because it implies that the food is unusually good.
However, if nobody prompted me, and I simply want to proclaim that the food I'm eating is good, I'd be more tempted to use the word "delicious" because it's specific to food, so it's more easily understood what I'm talking about.
As for "yummy," "tasty," and other such specific words... it's hard to say if they're common. I rarely hear "yummy" but I hear and say "tasty" quite often. Definitely more so than "delicious."