As Soon As, Just After, Immediately After, Quickly, Early
noun, suffix
Essentially you have the kanji 早 twice, because 々 is just a repeater kanji. When you have something early twice, it obviously has some kind of urgency to it. You'll need to take an action as soon as you finish whatever it is you're doing there.
そうそう, はやばや
The reading is just the on'yomi readings (which you learned with the kanji) twice in a row.
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.