Past repetition vs continuation or addition
又 strongly signals "this happened AGAIN, past tense": 他昨天又迟到了 (he was late again yesterday), 你又来了 (you came again). The repetition is a completed past event, often with mild annoyance or surprise. 还 covers three related but distinct meanings: "still" (他还在北京, he is still in Beijing), "additionally" (我还要咖啡, I want coffee too), or "yet more" (还有一个问题, there is one more question). 还 does not imply past repetition.
他还在那里,但又看错了时间。
Tā hái zài nàlǐ, dàn yòu kàn cuò le shíjiān.
He is still there, but he misread the time again.
又 = past repetition or established pattern
又 marks events that already happened and are now happening again: 又下雨了 (it is raining again), 他又忘了 (he forgot again). The "again" is observed, completed, often carrying a slight emotional tone (frustration, resignation, amusement). 又 rarely works for future repetition; for future, use 再 (see zai-vs-jiu comparison). 又 is reserved for "this has happened again, and I am noting it now".
今天又堵车了,每天都这样。
Jīntiān yòu dǔchē le, měi tiān dōu zhèyàng.
Traffic is jammed again today; it is like this every day.
还 = continuation ("still")
还 with a verb means the action is still ongoing: 他还在工作 (he is still working), 我还记得 (I still remember). The state has not changed since some reference point. Negated: 还没吃 (has not eaten yet), 还不知道 (still do not know). This is one of the most common uses of 还 in daily Chinese and pairs tightly with progressive or resultative verbs.
还 = additional ("also / more")
还 before a verb can add a new item: 我吃了米饭,还吃了面条 (I ate rice and also ate noodles), 你还要什么 (what else do you want?). 还 here means "in addition". In negation: 还没 (not yet, see previous section). 还 attaching "also / more" items is common in orders, lists, and accumulating information.
我喝了茶,还喝了咖啡,现在还想喝水。
Wǒ hē le chá, hái hē le kāfēi, xiànzài hái xiǎng hē shuǐ.
I drank tea and also coffee, and now I still want to drink water.
Future repetition = 再, NOT 又
For "will X again (in the future)", use 再: 我明天再来 (I will come again tomorrow). 我明天又来 sounds wrong because 又 marks past repetition. This split is a classic HSK 3 exam trap: same verb ("come again"), different tense, different adverb. Lock in: past repetition = 又; future repetition = 再; continuation or addition = 还.