What other possible ways of saying "I'll let you know"? I heard people in Northern Ireland say" I'll give you a shout", but I'm not sure it has the same meaning.
I'll let you know
us usually used to tell someone they'll keep you updated on a decision or situation.
Other phrases that can be used are
I'll tell you as soon as I know
I'll keep you updated
I'll keep you posted
Your second phrase
I'll give you a shout
usually only means someone will contact you regardless of the situation and may or may not have the same meaning as "I'll let you know", it would depend on context and custom.
I'll give you a shout and let you know when I find out.
I'll give you a shout to arrange meeting up later this week.