You should almost always make suggestions that include one or two cards in your hand, especially after you know one or two of the real murder components. (Though this might change with the number of players. I'm used to playing with 3 or 4 people.)
The end goal isn't to know who holds which cards, it's to know which cards are not held. The most efficient way to gather information isn't when someone shows you a card, it's when they pass; especially when all other players pass. This does give information to other players, but more to you.
Let's say the real solution is Prof. Plum with the Rope in the Conservatory. It's Player A's turn and he's in the Conservatory. If A makes a random guess of cards he doesn't hold, maybe A gets "lucky" and accuses Prof. Plum (correct) with the Revolver, and of course the Conservatory. What A learns is who holds the Revolver. If he's lucky it won't be the person on his left, so maybe he'll see a pass or two, but it won't be everyone. However, if A suggests Ms. White (in hand) with the Knife (in hand) in the Conservatory, everyone will pass, and he knows that the Conservatory is the room! The other players will know that they desperately need to check out Ms. White and the Knife and the Conservatory.
Now, let's step into another player's shoes. Player B knows that Player A learned something big, so she runs to the nearby Billiard Room and suggests Ms. White with the Knife. If anyone shows her a card, Player A knows that it's the Billiard Room. Great! Player A learned as much as Player B did from her own suggestion. If people pass until Player A, he shows her, say, the Knife, and A knows that either B or someone sitting to his left before B holds the Billiard room. Player A should now preferentially show the Knife rather than Ms. White to everyone.
Contrast this with Player B being a little more conniving: she decides to test Player A's knowledge more gradually. She accuses Col. Mustard (in her hand) with the Knife in the Billiard Room. Now if someone before Player A shows her a card, Player A can't be sure if it's the Knife or if it's Col. Mustard. If it gets around to Player A, he doesn't have a choice of what to show Player B, he must show the Knife, which is also better for B than if A gets to choose.
As for suggesting 3 cards from your hand, that can be a devious way to throw people off track, but you also essentially lose a turn of information gathering so I wouldn't recommend doing it often.
Your card distribution (especially what rooms you hold) can affect your ability to make suggestions involving cards from your hand. I try to make one of the first 3 rooms I visit a room I hold. It's especially nice if you hold two secret passage rooms to hop back and forth between the two (though pretty soon everyone will know what you're doing). It can be tough if you hold only one card (0 even worse!) of a category. I'll focus on that category first so that I can at least alternately suggest 2 items when I get to the other categories.
In the beginning of the game it's fine to make suggestions with only one card you hold, or the occasional 0-held-card suggestion. You'll learn something--maybe you'll get lucky and learn a lot. But as soon as you know one category there's no point in making a suggestion involving a card in that category that someone else holds. You already know somebody holds the card, but in most cases you shouldn't care who.
tl;dr You learn more from people passing, and people pass more often if you make suggestions that use some cards from your hand.