I saw that there was not much water left in the bottle, so I picked it up went to the kitchen refilled it.
There is a problem with this sentence: you need to add commas.
I saw that there was not much water left in the bottle, so I picked it up, [I] went to the kitchen, [and] [I] refilled it.
In very casual, informal writing you might leave out and, but it is normally compulsory. Leaving out I in the second and third clauses is perfectly normal and not informal at all. There is nothing unusual about to: it just introduces the direction of went, an adverbial phrase of direction.
I picked up the bottle [in order] to refill it.
This is perfectly normal. You don't need in order: it means exactly the same thing. In a simple, clear sentence like this, using to to introduce a purpose with to + infinitive is standard. It has nothing directly to do with to as in the first sentence, where it introduces a direction.
I saw the show to talk about it next morning at work.
This sentence is of the same type as the previous one: the reason/purpose why you saw the show is that you could talk about it the next morning. The situation is a bit unusual (is that really the main reason why you saw the show? Isn't that a bit silly?), but the grammar is fine.