These adverbs could be used to indicate the recurrence of an action or the continuation of a situation. But still, there are some differences among them.
Yòu (又) usually comes before a verb, indicating that the re-occurrence of an action has already happened, therefore in most cases, it refers to events in the past .
tā zuó tiān qù kàn diàn yǐng le, jīn tiān yòu qù le.
He went to the cinema yesterday, and he went there again today.
The use of Yòu (又) sometimes indicates predictable repetitions, that a regular, periodic action or state which has happened before will happen again in the immediate future.
míng tiān yòu shì xīng qī yī le, yòu yào shàng bān le.
Tomorrow is Monday again, and I'm going to work again.
The use of zài (再) usually indicates that the recurrence or the continuation of an action has not yet happened, but will happen in the future.It is noteworthy that when there is an auxiliary verb in the sentence, zài (再) should be placed after the auxiliary verb.
zhè běn shū tài hǎo kàn le, wǒ yào zài dú yí biàn.
This book is so great, I'll read it again.
Hái(還) signifies the continuation of an action or state, which often occurs in sentences with an auxiliary verb. Hái(還) could be used together with zài(再), in this case, the auxiliary verb should occupy the position between hái(還) and zài(再).
zhè běn shū tài hǎo kàn le, wǒ hái yào (zài )dú yí biàn.
This book is so great, I want to read it again
The use of Hái(還) often implies someone’s subjective desire or intention for the action to be repeated, so it often collocates with yào (要) or xiǎng (想). Unlike hái(還) , zài(再) emphasizes an objective need for the action to be repeated.
wǒ míng tiān hái yào lái zhǎo nǐ, gēn nǐ liáo tiān tài yǒu qù le.
I'm coming to see you again tomorrow, it's so interesting to chat with you.
wǒ míng tiān zài lái zhǎo nǐ, jīn tiān tài wǎn le.
I'll see you again tomorrow, it's too late today.
The adverb zài cì(再次) conveys the meaning of once again or for a second time,which usually precedes disyllabic verbs. Unlike zài(再), zài cì(再次) could used be in past tense sentences.
tā jīn nián zài cì huò jiǎng.
He won another prize this year.
hěn gāo xìng zài cì jiàn dào nǐ.
Nice to see you again.
Zài sān(再三) can be placed before or after disyllabic verbs, which means “repeatedly” in a formal way.
kǎo lǜ zài sān, wǒ jué dìng fàng qì zhè gè jī huì.
After much consideration, I decided to give up this opportunity.
lǎo shī zài sān qiáng diào, kǎo shì bù xǔ zuò bì.
The teacher repeatedly stressed that cheating is not allowed in examinations.