As Soon As
Immediately After, Quickly, Early
noun, suffix
Essentially you have the kanji 早 twice, because 々 is the repeater kanji. When you do something doubly early, there must be some kind of urgency to it, right? So you need to take action as soon as you finish whatever it is you're doing.
This word has two possible readings, each with a different nuance, so here are some mnemonics to remember them by:
そうそう is for when you do something quickly or immediately after something else, like 早々にホテルを出る (to leave a hotel quickly) or オープン早々 (as soon as a shop opens). You can remember this because as soon as you get to work, you do everything really quickly, so your work performance is just so-so (そうそう).
はやばや is for when you do something earlier than usual, like going to bed or closing up shop before the usual time. It often appears as 早々と. You can remember this because when you want to close up shop early, you say "Hiya… bye-ya!" (はやばや) to all the customers who come in. Saying both hi and bye when they walk in is the best way to hint that you're trying close early. Hiya, bye-ya! Yeah, they get the hint.
早々にメールありがとうございます。
Thank you for your prompt email response.
ホテルを出て早々アパートをかりた。
I started renting an apartment just after leaving the hotel.
うどんパーティは、つまらなくて早々と引き上げちゃった。
The udon party was boring, so I left early.