One of the most common mistakes in lettering is found within most styles of contrasted type: where people use thick and thin strokes. It's a dead giveaway showing that you don't truly understand typography. If the stroke weight isn't in the right places, your lettering will feel unbalanced. The main culprit (and my biggest pet peeve) is the capital A.
Look at the difference between the two in the image above. Something just feels off in the second one, doesn't it?
The simple rule is this: downstrokes are thick, upstrokes are thin. Think about how a letter is actually written to determine where the thicks and thins go. So in a capital A, you draw from the bottom left diagonally upwards and right. That's a thin stroke. From there, you then draw diagonally downwards and right. That's a thick stroke.
Let's dive into the details a bit more.
Shocking as it may seem, computers weren't always around. Throughout the centuries, there were different writing utensils and different styles of writing. Often you would find some sort of chiseled or flat tip on the tool: in the ancient Roman days, it was literally a chisel used to carve into stone; in the Victorian days, it was a fountain pen.
Those tools result in varying stroke widths within each letter. The "proper" way to write is to have a thick stroke when the tool is moved in a downward motion and a thin stroke when it's drawn upward. (I put "proper" in quotes because rules are meant to be broken. There are plenty of funky fresh typefaces where the complete opposite occurs: thin downstrokes and thick upstrokes. But approach with caution! You have to learn the rules first before you can break them.)
Basically, no matter what tool you're using, think: downstroke = thick, heavy, more pressure.
But what about horizontal strokes (ex. A, E)? Those are thin as well. Diagonals? Depends on the letter. (ex. M, V). Think back to elementary school where you had those worksheets with the dotted lines and arrows on how to properly form each letter. Those arrows tell you where to use thick or thin strokes based on the direction your writing utensil moves. So the next time you form a letter, pay attention to which direction your writing utensil is going. Up (and right) will be thin strokes; down (and left) will be thick.
You don't need to have special chisel tip or brush pens to get the thicks and thins. You can use a pencil too. A simple exercise when using a pencil is to apply more pressure on the downstrokes and less on the upstrokes. It will give the illusion of a thicker downstroke since that line will be darker. Then you can go back and thicken it to get the finished look.
And of course make sure you practice, practice, practice! Even though you may be focusing on lettering, having basic knowledge of calligraphy (i.e. thicks and thins) will greatly improve the overall quality of your work.