I usually spot a sentence like this in books or other social media tools and try to apply it in my writing and speaking without understanding how it works =)) It would be appreciated if you guys can give me an explanation.
Here it is: "He stands still, watching us argue."
And here is what I formed myself: "I wanna get out there looking for some ***", is it right?
You have an ambitious and experimental approach and that is good. At your level, I recommend that you try to understand the purpose, the aim of each grammatical construction. For this, you probably need to study more.
In the first sentence, "watching us argue" is an adverbial phrase. The phrase describes "He stands still". It tells us more about how "he stands still". The meaning of the sentence is similar to "He stands still and at the same time, he watches us argue."
Here is a similar example: "Driving his car down the road, my friend waved at me." You can see that adverbial phrases like these start with -ing.
The first sentence has a literary style and so copying it in speech is not likely to sound normal.
In your own sentence, the function of "looking for some ice cream" is not clear. My guess is that you wanted to describe your purpose, the reason why you "want to get out there". The easiest way to do this is with an infinitive. e.g. "I want to get out there to look for ice cream."