sample.vue
//省略
<script>
export default {
    //省略
    methods: {    
        method1(){
            this.$axios.$get('URL1',{params: {param1: param1, param2: param2})
            .then(response => {
                //　最初の処理
                this.item1 = response.item1
                this.item2 = response.item2
                //if文も可能
                if(response.data != null){
                    this.data = response.data
                }
                return this.$axios.$get('URL2');
            })
            .then(secondResponse => {
                // 2つ目の処理
                return this.$axios.$get('URL3');
            })
            .then(thirdResponse => {
                // 3つ目の処理
            })
            .catch(error => {
                this.message = error
                //errorも複数行可能
            })
            .finally(() => {
                //hogehoge
            })
        }
    }
}
</script>

<script>
export default {
    //省略
    methods: {   
    // async awaitを使用する場合
    async getData(){
        try{
            const response = await this.$axios.$get('URL',{params: {param1: param1, param2: param2})
            if(response != null){
                this.item1 = response.item1
                this.item2 = response.item2
            }
        } catch (error) {
        // handle the error
      }
    async getSomeData() {
      try {
        const [response1, response2] = await Promise.all([
          this.$axios.$get('/api/v1/example/1'),
          this.$axios.$get('/api/v1/example/2')
        ]);
        console.log(response1)
        console.log(response2)
      } catch (error) {
        // handle the error
      }
    }
   }
}
</script>