I know Korean has formal and informal ways of speaking, but is there a complete difference between formal and informal speaking or just small differences? Say I was speaking to a person older than myself, would I say Hello or How are you formally then go back to speaking informally, or do you speak everything formally?
Korean has formal and informal embedded in the grammar, much like most European languages with a T-V distinction but with more levels. Grammatically, there are quite a few "speech levels", from archaic Joseon-dynasty / Biblical [하소서체] to casual between friends or to children [해체]. However, I think most beginner Korean courses teach just two: 해요체 and 해체, with some 합쇼체 added later, as these three are ones you will come across daily in modern South Korea.
Native Koreans split the system into 존댓말 for "formal" vs 반말 "informal", which in modern South Korean society grammatically really means a mix of 합쇼체 and 해요체 for the first, and 해체 with a little bit of 해라체 for the second.
Also, I consider it more useful (especially from a pedagogical level) to break down Korean's system into three dimensions: formality, politeness, and honorificity. It is possible to be informal and yet polite [해요체], or formal and non-polite [해라체].
How the relationship of 존댓말 to 반말 is established is really important. In general, professional environments would mandate 존댓말 all the time, even with people you know in a different context. In many professional contexts, including (famously) men in the military, even 해요체 is disallowed (합쇼체 is required in all formal contexts); also most news broadcasts and official announcements.
In most daily conversations with people you know, 해요체 is most common, and as the younger one in the relationship you would (in general) not be at liberty to change your speech level; that would be the right of the older one. Hence why most Korean speakers ask for the age when meeting someone for the first time.
In the event that the older one in the relationship wants to switch from 존댓말 to 반말, the older one would initialise, but it can be a clear declaration, or they can go straight into using it, or they can mix between 해요체 and 해체 for a bit. The younger one may not necessarily reciprocate with 해체 back. That's quite common in fact. E.g. the first relationship i.e. between parents and children, the speech levels would be 해체 from parents to children and 해요체 from children to parents, although that is changing somewhat in modern society.
It is possible for the younger to encourage the older one to "lower their speech" [e.g. by saying 말 놓으세요 or 말 편하게 하세요], but note that younger will often (/ should) still stick to 존댓말. This is particularly common within structured but informal environments, e.g. to a 선배 at university or within the workplace.
If you are both the same (Korean) age, there is already a basis for using 반말, and some people will go for informal straight away, but not in all cases.
Hence, apart from clear cases of 존댓말, there is a bit of flexibility in managing the transition from 존댓말 to 반말. It's a big part of K-drama plots: the point at which the relationship goes to 반말, whether it's one-sided or mutual 반말 etc.
I've not mentioned the drop from honorific to non-honorific within 존댓말 very much, but that's also a subtly managed transition, and also as indicative of how someone feels about the referred-to person.