高兴 is not To be used in everyday life. I guess it's like that in English people say I'm in heaven rather than I'm happy. It's so ordinary and plain and boring that I don't use it Except in articles. If you say 我很高兴... ,it gives me a feeling of meaning "I'm gratified that you did so" from a teacher or an senior person praising me. You can't say 我很愉快，you can say 愉快的心情 , it's not wildly used when talking to friends, I guess. 我很开心 is better than 我很高兴，or “开心的一天”，but still boring. I don't even think you are really enjoying yourself when you use it.=_= I would use 太爽了！！ 爽爆了！！to express happy mood. Anyway, when really into an extraordinary mood, don't use all three of them.

高兴 is more happy then 开心, 愉快 means haapy but do not show it. if someone feels 高兴，maybe he jumps, or dance, or laugh out , or smile, if someone feels 开心，maybe he smile, but not laugh out or dance and so on. if someone feels 愉快，maybe you do not know. he just feel happy or feel very very hapy in his heart at all.

基本一样，都是表示happy的意思。但是在用法上，开心和高兴比较口语化，常做表语，如：我很高兴，我很开心。 愉快比较书面化，常用作定语，形容一段时光，如：我度过了一个愉快的周末。愉快表达happy的程度比较轻，侧重于轻松、放松的意思。

愉快 ;心情开心 高兴:愉快，兴奋 开心，汉语词组，意指开露心意、坦诚相待，或者心情舒畅、快乐等。