The truth is, English learners tend to avoid these two punctuation marks because they are uncertain about their usage. Take a look at them:

Colon (:)

1. Explanation

A colon usually introduces an explanation or further details.

• We decided not to go on holiday: we had too little money.
• Writing an assignment is not easy: to begin with, you have to do a lot of research.
• A woman: without her, man is nothing.

2. Lists

A colon can introduce a list.

• Buy these things: a packet of peanuts, two loaves of bread and a kilogram of steak.

3. Letters

Americans usually put a colon after the opening salutation (Dear … ) in a business letter.

• Dear Mr. Callan:

Semi-colon (;)

1. Instead of full stops

Semi-colons are sometimes used instead of full stops, in cases where sentences are grammatically independent but the meaning is closely connected. Semi-colons are not nearly as common as full stops or commas.

• Some people work best in the mornings; others do better in the evenings. 
• It is a fine idea; let us hope that it is going to work.
• Don’t go near the lions; they could bite you.

2. In lists

Semi-colons can also be used to separate items in a list, particularly when these are grammatically complex.

• You may apply for a visa provided that you have a passport; that you fill out necessary forms; that you ensure your return to your home country; …

So, don’t be afraid to use these two handy punctuation marks next time you write. 🙂