English is very full of article usage (a, an, the). The general rule in English is to use "the" when you are talking about a specific subject (so you would go to THE movie, not just "movie)--"the" is called the "definite article." If you wanted to go to the cinema but you did not know which specific movie you wanted to see, you would say that you want to go to "a" movie--"a" is an indefinite article. If you do not use either "a" or "the" in this sentence, English speakers will still understand you but you will not be speaking correctly.
For your second example ("going to market"): In American English, we will always say "I am going to a market/the market"--we do not leave out the article. However, I believe that English speakers in other parts of the world (like England, for example) might omit "the/a." These people would then just say "going to market." I do not know why this difference exists. It could be that, because true outdoor markets are much rarer in America, people will not always understand what we are talking about if we were to just say "going to market" (because for most Americans, there isn't a market to go to!).
I hope this helps!