How can I say basic phrases "Do you mind if I speak in Chinese?" "Feel free to correct me" "How do you say (X) in Chinese?"
Do you mind if I speak in Chinese?

Technically this could be translated to e.g.:

我说汉语，你不介意吗？
Wǒ shuō hànyǔ, nǐ bù jièyì ma?

This literally means &quot;I speak Chinese, you not mind?&quot;  However, a beginner will likely need a lot of study before being able to pull this off.  Far simpler is &quot;Let's speak Chinese&quot;:

我们说汉语吧。  [Let's speak Chinese.]
Wǒmen shuō hànyǔ ba.

Next:

Feel free to correct me

I don't know how to say this off the top of my head, and I've been studying Chinese for years.  After a bit of Baidu-ing, this seems close:

随时地纠正我说的。 [Correct what I say at any time.]
Suíshí dì jiūzhèng wǒ shuō de.

I changed the meaning to &quot;Feel free to correct what I say&quot; (so it excludes e.g. &quot;correct me&quot; in the sense of e.g. correcting posture, mannerisms).  In any case, it's going to sound like English translated into Chinese, and not real Chinese, and will likely just result in blank stares.  More natural is:

请指出我的错误。 [Please point out my mistakes.]
Qǐng zhǐchū wǒ de cuòwù.

(If we Baidu search for 指出我的错误, we can see it's used.)  Or to be a bit more precise:

请指出我说的任何错误。 [Please point out any mistakes in what I say.]
Qǐng zhǐchū wǒ shuō de rènhé cuòwù.

Finally:

How do you say (X) in Chinese?

There are many possibilities, such as:

你怎么说&quot;Australia&quot;？  [How do you say &quot;Australia&quot;?]
Nǐ zěnme shuō “Australia”?
&quot;Australia&quot;汉语怎么说?  [How do you say &quot;Australia&quot; in Chinese?]
&quot;Australia&quot; hànyǔ zěnme shuō?
这个用汉语怎么说?  [How do you say this in Chinese?]
Zhège yòng hànyǔ zěnme shuō?