Check out the two Wiki links below. They explain and show clearly that the crest is one small component of a coat of arms; it represents the crest that armoured knights used to wear on top of the helmet as additional identification. When a full coat of arms (the correct term is actually "an achievement of arms", but only heraldry geeks know that), is shown, the crest is always depicted on top of the helmet, which itself is always shown above the shield. These days the shield is usually shown upright, but in older art it's often shown on a slant with the helmet balancing on one corner - this represents the way a real shield and helmet would be hung on a wall when not in use. You can see a genuine medieval shield and helmet with crest hung up in this way in the third link.
You can only have and use a crest if you have a coat of arms of which it is a part. The confusion between "crest" and "coat of arms" arose:
(a) because people used to put their coat of arms on their belongings, which in a pre-literate society was a practical way of marking them. But unless the coat of arms was a very simple one indeed, it just wasn't practicable to engrave it on really small items; so the practice arose of using only the crest, sitting on the "torse" or wreath that in the whole arms covers the join of crest and helmet, on small things like coffee spoons and signet rings. So you quite often see crests being used on their own; and many people don't realise that here the crest is just a part representing the whole, and believe that a crest can exist independently.
(b) And crests used in this way have themselves got confused with heraldic *badges*. Medieval lords mostly didn't give their followers uniforms - they would give them a badge to wear to show who they belonged to. It wouldn't be the whole coat of arms (only the lord himself wore that) but it usually was a component of it. These days hardly anybody has feudal followers except Scottish clan chiefs, and it is their chief's badge that clan members are entitled to wear and display - not his crest, and certainly not his coat of arms.
So, to your problem: how do you find your coat of arms? Well, you must first realise that the chances are that you don't have one. Only a very small minority of people do. Coats of arms were only adopted by the upper tier of society - people with property and social standing. In the British Isles, only people of "gentry" status had them - in fact inheriting or being granted a coat of arms was explicitly a proof of this status. In some other countries (e.g. Germany) the habit of adopting arms reached down to people with a more modest standing but still people with property and importance in their community - e.g. merchants who owned their own business, farmers who owned their land. Working class people didn't have them, and the fact is that most of us are descended from working class people.
However, if you want to try anyway, there is no short or cheap route: you have to research your ancestry back, in the direct legitimate male line, to someone who did have a legitimate coat of arms. Then present your research to the heraldic authority of that country, and ask them to verify your right to that coat of arms - or, more likely, a "differenced" version of it, if yours is not the senior line of the family. (In Western European heraldry only one person is ever entitled to use the basic coat of arms, and that is the current head of the family. Everybody else - his children, his younger brothers, members of "cadet" branches - can only use a version specially altered to show that they aren't the head of the family, and what their relation is to him.) The heraldic authority will make a charge for this; it could cost quite a lot of money.
Good luck with this. Please don't get a tattoo of whatever you find on the internet claiming to be "your family's coat of arms": that's an expensive, painful and permanent way of making a fool of yourself.