Onyomi is used in jukugo(words with more than one kanji). Kunyomi can be used by itself. Note, there are many exceptions to this. I'll give some examples. Once you start practicing with kanji, you'll understand it with time.

教 Education　ON:キョウ　教える「おしえ・る」 To teach

室　Room ON: シツ

教室「きょうしつ」 Classroom

There are words were the onyomi is a word used by itself, but these instances can be memorised. When reading, it can be assumed it's the kunyomi. For example: 本 On:「ホン」 Book

There are other hints which reading to use. Okurigana are the kana written after a word. える is the okurigana in the word 教える. Many kanji have many kunyomi, the okurigana can be used to tell which word it is. When you see okurigana in a word, it's the kunyomi reading being used.

私「わたし」に教えてください。 The verb is conjigated to te-form, however you can still see the okurigana there regardless, so it's no mystery it's the kunyomi being used.

There are kanji where the kunyomi acts as onyomi and goes in jukugo. But there aren't very many, don't worry about them too much. For example, 夕 and 名.

This is just my two cents, I have had the same problem and this is just my personal solution. Hopefully my logic helps you.

I will use the common phrase 

"I love you" as an example.

When you read that you take a combination of letters and create words and they in turn create a sentence. It will always spell the same thing.

However what if it were a picto gram?

EYE HEART (followed by a) hand pointing at you

you could intrepret it many different ways, but some people would eventually understand that 

it could be i love you as well,

together they create a sentence that makes sense only together, while apart they could have different meanings.

essentially this is Kanji

話す＝hanasu to talk

話＝hana but because its accompanied by su you figure its a verb

while

電話＝denwa telephone or electric talk

話＝wa

its just practice, memorization and figuring where the pieces go, the more you remove English from the equation, the easier it becomes.

Hi,

not only you but also many Japanese children are working hard on Kanji.

In short, you can remember that "on"-yomi comes from Chinese. Not exactly the same but similar.

In Japanese we had words or phrases, but we did not have letters to write, more than 1500 years.

Buddhism Bible from Indea was translated in Chinese in China and then bourght to Japan.

Japanese people tried hard to understand the letters.

They somehow identified many (or almost all) letters from the meaning, and assigned comonlly used Japanese words to the corresponding letter. That is why we have some ways to read one letter.

The reading which somehow keeps the Chinese pronunciation is called "On"-yomi, which we write "Sound"-reading, so there is no meaning. Ariana san showed an example with 教. If we would say "kyo" nobody understands what it is only by that.

On the other hand, reading from original Japanese reading is called "Kun"-reading. This would make sense as soon as it has been spoken. I can also say, that it was how Japanese imported foreign letters and adapoted.

And as you might know, from Kanjis old Japanese developped Hiragana and Katakana.

There are some Kanjis which originated in Japan and then exported to China, such as "democracy", "culture" and so on, which were created in early Menji era just after Shogunate time to enable European culture or technology available in Japanese for Japanese students or citizens.