The Chinese name given to me by my teacher is Ma Jialing. How would I introduce myself?

You would introduce yourself as Ma Jialing. In Chinese, the proper way to introduce yourself or address someone is by their full name (Last and then First).

People can call you by your first name only when you've become closer (ie. friends, family) or if it's an elder calling someone younger (ie. an eldery calling a young volunteer at the old folk's home).

Ma (surname) Jialing (name), Chinese people say surname and name together, don't mind the singers, unless you aren't yourself...
in China people never call anyone by first name, only full name or Mr/Mrs surname...

Also, even for people you meet the first time in a casual setting, it is still polite to introduce yourself as Ma Jialing and then offer that they can simply address you as Jialing.

In Chinese names, the family name always comes first. Only when used in a western context, and especially so if used together with a western name (Faye Wong) is the family name used last. In my experience most non-Chinese and Westerners would be easily confused if you wrote your name as Ma Jia Ling they might think the family name was Ling.

Finally, it is common in multi-lingual Chinese cities like Hong Kong, Singapore, Taiwan etc., to use both Eastern and Western names together, in which case - Faye WONG Fei, Bruce LEE Xiao Long, Jackie CHAN Long.

Tomato SAUCE Is Good (just kidding).

friends will usually call you by your first name. only in a bit more formal occasions will people call you with your first name. So when you meet people, if they are professionals you should refer yourself as ma jialing and if the occasion is casual, use jialing.