1. When you study kanji, learn them by words.
For example, suppose you want to learn 大.
It's not efficient to memorize mechanically like "kun reading is 'oo' and on readings are 'dai, tai'...." Isn't it boring?
Instead, I recommend you learn words such as "大きい(おおきい)" ,"大人(おとな)", "大学(だいがく)" . In this way, you don't really need to even care about if it's on or kun. (That's what Takako means in her first line, I think.)
When Japanese pupils take kanji test, it looks like this:
Q. Write the underlined kanji in hiragana.
"わたしの家は 大きいです。"
ー
A. (おお)
2. You may come across with unfamiliar words written in kanji, and need to know how to read it. Here are some hints.
-If it comes with okurigana, it's most likely kun reading.
-Kun reading makes sense as a word by a single kanji itself or with okurigana if any. Most on reading doesn't make sense by itself.
-It the word is made of two or more kanjis in row, it's likely to be on reading by 70-80% possibility.
-To be 100% sure, get a electric dictionary, go to some kanji website, or use IME on PC that allow you to input by pen/mouse writing. It will give you the answer.
大きい→大is alone with okurigana. So it's おおきい。
大学、大統領、公明正大→Two or more kanjis in row. So they are probably on reading. And yes, they are. だいがく、だいとうりょう、こうめいせいだい。
大空→Two kanjis in row, but this one goes into the minor 20%. It's おおぞら.
大家→Could be kun(おおや) or on(たいか) depending on the context. You really need to know the meanings first. May need to look up dictionaries to make sure.
*Okurigana
大きい→きい is okurigana.
白い→いis okurigana.
食べる→べる is okurigana.
飲んだ→んだ is okurigana.
Note that する for verb, な, だ/です for nouns and na-adjectives, and their variations are not counted as okurigana for this purpose. For example, 元気な looks like it has okurigana, but it's read by on reading げんきな.