How to say something like "Let's go [do something]"
For example, what if I wanted to say, "Let's go to see a movie this Friday."
How can I say the "let's go" part?
In addition to Karen's answer, the casual speech of しよう is conjugated differently for each verb depending on the ending. Here are some examples.
行く、行こう iku, ikou
見る、見よう miru, miyou
泳ぐ、泳ごう oyogu, oygou
する、しよう suru, shiyou
飲む、飲もう nomu, momou
座る、座ろう suwaru, suwarou (this one is different than 見る because 見ます and 座ります are conjugated differently)
Basically, if the verb ends in something other than る just change that hiragana up to the next last one of that family.
For example, 行く ends in く. So if we look at that family か き く け こ we can see that こ is the last one. We add う and that makes 行こう. It takes a little getting used to, but you'll get them down with practice in no time.
You can say something similar to "let's" by adding ましょう to the end of a verb for polite speech, or simply しよう for casual speech with your friends.
Let's go to see a movie this Friday.
今週の金曜日に映画見に行きましょう。konshuu no kinyoubi ni eiga wo mi ni ikimashou.
金曜日、映画見に行こう。 - kinyoubi, eiga mi ni ikou.