Which expression do native speakers actually use: Chinese New Year or Spring Festival (or other)? In relation to the Chinese New Year (or Spring Festival) I've heard different names, as you can see from what I just wrote.
In Chinese we have:

春節 / 春节 = Spring Festival
Chūn jié
農曆新年 / 农历新年 = Chinese New Year
Nóng lì xīn nián

or simply (more general)

新年 = New Year

Apart from the last one, which I doubt is the most used when referring specifically to the Chinese New Year, which one is the most common expression in China? And are these the only ones?
The most common expression should be 春节.

农历新年 is a kind of rare.

We have another word for January 1st (not spring festival), 元旦(yuan2 dan4). So 新年 is normally the Chinese New Year.

Another expression is 阳历年(yang2 li4 nian2) for January 1st, 阴历年(yin1 li4 nian2) for spring festival. 阳历年 may be used more often, as 阴历年 is the default.