Example sentences
xīn新nián年kuài快lè乐！
Happy New Year!
tā他guò过dé得hěn很kuài快lè乐。
He lives happily.
About this word
Both halves already lean toward joy before they are joined. 乐 is the character for music, and Chinese lets one word cover both music and gladness; the same character reads yuè for music and lè for joy. 快 normally means fast, but that is the newer sense. Its older meaning was pleased and cheerful, which still shows in 愉快 (cheerful), 痛快 (delighted) and 大快人心 (to the great satisfaction of everyone). Combine the two and you get a plain coordinated compound, happy, carrying about the same weight as 高兴. 快乐 shows up more in set phrases and good wishes, as in 生日快乐.
The story behind the character
快
A surprise here. 快 is 忄, the heart, with 夬 for the sound, and Xu Shen defined it 快, 喜也: pleased. The heart radical is the clue. 快 spent its early life meaning happy and delighted. Fast came along later, perhaps from the sense of something pleasing arriving quickly. 快乐 keeps the old meaning.
乐
乐 began as music. The traditional 樂 is a wooden frame (木) with silk strings (糸) above it, a zither drawn in pieces. Xu Shen saw a drum on a stand instead: 乐, 五声八音总名, the general name for all sound. Music meant pleasure, so yuè slid into lè, happy. 快乐 still has wood at the bottom.