再 means "I will do it again in the future." example : 再见 (See you later) 又 means "I did it twice/again." 一边 means simultaneously. Most of the time, we say "一边...一边..." example: 他一边看电影，一边吃爆米花。

再和又各自的多种义项里只有都表示again时容易混淆。其实，英语里的一个again在现代汉语口语里用「再」和「又」两个词表示很大程度上是为了弥补汉语动词没有时态的缺陷。简而言之，「再」大致对应将来时，「又」大致对应过去时。比如「我今天去了市场，明天再去。」「我前天去了市场，昨天又去了。」如果这两个句子的「再」和「又」互换明显不可接受。另外，「再」还有「现在没做将来做」的含义，比如「今天没去成，明天 再去吧。」 作者：王伟超Mijiag 链接：https://www.zhihu.com/question/25575521/answer/33681897 来源：知乎 著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。

再 and 又 I think David explained very well，and 一边 emphasises the meantime.

"Zai" is used for something that will happen again in the future. "You" is used for something that happened again in the past. I can't explain the other one (sorry, I'm just learning too).