What are the recommended Chinese songs for Chinese learners to learn

You can try 王力宏's songs.

The Chinese edition of “Auld lang syne” is called 《友谊地久天长》, you can search it to listen.

我觉得最好还是和国外有关联的歌曲，例如小星星和生日歌

If you can access YOUTUBE, there are lots of Chinese songs, depending on what taste you have.

To start with children's songs.  

Besides rap songs, every Chinese song will help you become familiar with Chinese

songs of Teresa Teng can help learn chinnese. Listening to her songs, you can hear clear Chinese pronunciation. At the same time, some songs have simple lyrics and short sentences. Some of the songs have Chinese connotations.recommended《甜蜜蜜》and《小城故事》
 

Chinese children's song if in elementary level (两只老师，数鸭子）

listen to Jay Chou's music if your level is higher than that , his songs are more Chinese style (东风破，青花瓷）

You can try Kugoo music

You can check QQ music and find the most popular songs in China. They are catchy!