So I’m new and excited about learning Japanese.
I’ve learned the Kanji for big 大 (たい) and person 人 (にん) so when I saw a sign with 大人 I logically went “big person” and was stocked that this sign was referring to an adult.
So why then is the Japanese word for adult not たいにん? I’ve looked it up and the word is otona??
Any help in understanding would be appreciated.
Even though it is a jukugo (kanji compound), 大人 doesn’t follow any rules, so you just have to learn the reading separately. Unfortunately, this will come up quite a lot in Japanese, where the reading doesn’t seem to follow any rules, so you should get used to it. Another example for this is 今日.
There are also jukugo that don’t use the on’yomi readings of the kanji, but the kun’yomi readings, like 大空, then there are also words that mix on’yomi and kun’yomi readings, like 辛子, and then there is also rendaku.
I recommend having a look at this to get a better understanding of all this: Onyomi vs. Kunyomi: What's the Difference?
Hm, I’m no etymologist so I could be way off base here, but my gut feeling is it’s this way because spoken language comes before written. Since kanji are Chinese characters, some concepts won’t align with the spoken language. Again this is just unresearched speculation on my part 
Not that this helps learning them, but might lessen the frustration.
I once read that when Japanese was spoken, not written, they came up with certain words, including Otona (I’d imagine). Then, when Chinese characters came along they just slapped the best-fitting kanji on their pre-existing words. So just relax and try learn them 
I see @noko 's comment and yeah, good instincts! Unresearched speculation for the win!!
The technical term for the readings of words like 大人 and 今日 is 熟字訓. It’s a reading which belongs to the assemblage, but which doesn’t come from any of the components.
(Not so much for 日本 or 一人 - those are just slightly less-common kun’yomi readings.)
I feel like 日 is going to kill me one day with all the readings it has 
Japanese has broadly speaking two categories of words: ones which derive from the old Japanese language as it was spoken before the writing system was invented; and words which are originally loanwords from China. (This is a bit of an oversimplification but it’s a helpful place to start.)
The loanwords from China came along with the kanji used to write those words, and the Chinese pronunciation of the time, which got a bit mangled because Japanese doesn’t have the same set of sounds Chinese does. These are the words written with on-yomi. (Newly coined words like words for scientific terms added in the 19th century also often are on-yomi compounds.) If you see one of these words but don’t know the reading for it, you can often guess it based on the readings of the kanji in it.
But for writing, people also needed a way to write the words Japanese already had, so for those they kept the sound (obviously) and used whatever Chinese character or characters had an appropriate meaning. These are the words written with kun-yomi and with the just-for-this-word 熟字訓 that Belthazar mentions. They’re often the more common and conversational words. You usually cannot guess the right reading for these words, you just have to know them.
So in this specific case, the word for adult is in the second group, and its reading isn’t a simple “glue the readings for two characters together” one.