Hello. Your question is an important one -- many of my students find it difficult to see the difference between them. The difference is that "first" is used when we are making a list, as in "There are three places in Japan that I want to visit. First, I want to visit Tokyo. Then I want to go to Kyoto. Third, I want to visit Osaka." As you can see, "first" is used for "the first item" in a series.
On the other hand, the phrase "at first" is used to show a contrast between one situation and another. For example, "At first, I wanted to go to Tokyo, but my friend said it was not so interesting, so I went to Kyoto instead." In other words, "at first" suggests a situation that later changed. That is why it is often followed later in the sentence by "but" or something similar. Here are another couple of examples: "At first, I didn't like sushi, but now I love it." "At first, I felt too hot, but then I cooled down and began to feel more comfortable."
Anyway, Dongxiaoyu, I hope that the situation is clearer now! When you have questions about English, you can do several things. "First, come to UsingEnglish.com (then go to other sites)!" When you are learning English, there may be many puzzles about how to use words. "At first, it may be unclear how to use words, but if you study hard, it will become clear!" Good luck.
Rick S.
