You should be aware that your problem is a little more complicated than Jomo's example, though his advice is exactly what I do.
Since you need at least 3 cards of the same suit, there may be exactly 3, 4, 5, or 6 of one suit. Unless you see a foolproof shortcut, I'd calculate each of those four cases separately. The first is perhaps the trickiest, because your count may double-count a case with three of one suit and three of another, since it would be counted for each of those two suits. The others should be easier, so maybe you should start with the easiest case: How many possible hands contain 6 cards of the same suit?