A more formal construction, "ni atatte" is used when you are marking the occasion. "Ni saishite" has a similar nuance, think of both as a fancy way to say "when."
You can tack "tai" on the end of a verb to express what you want to do, but if you're going to speak for someone else then it's best to get a handle on the "garu" ending.
The structures "nagara ni" and "nagara no" are used in very specific ways in Japanese, which makes their comprehension slightly easier.
Using the grammatical structure "X kuse ni Y" has the same effect as "despite X, Y" and can be used to convey a feeling of ill will.
The "sae ... areba" pattern is used to express ideas like "if only" and can be used in several simple situations as well as more complex ones.
The particle "hazu" basically translates as "should" in English, but can sometimes be rendered as "no wonder."
The suffixes "zukume" and "zukushi" express the idea of "entirety" and "variety," and are sure to give you an edge in more formal situations.
The adverb "semete" can mean "at least" in English, but there's a nuance that goes with it that further specifies how it can be used.
The word "bakari" can be used in many situations in Japanese and can be seen in some of these cases as a replacement for the two different meanings of the English word "just."
Whether it's used to connect sentences or simply to add emphasis, "noni" is important to convey the idea of "even though."