Stepping in for my Chinese to Australian translations:
Also a couple of other phrases that are good to keep in your toolkit
Is the most basic and common way of saying thank you
谢谢你 Xièxiè nǐ
This is a more sincere or formal way of saying thank you
Translates to many thanks, Where 谢谢你 Xièxiè nǐ is used to more formally express a greater amount of thanks 多谢 Duōxiè is a slightly more casual way of saying the same.
Cheers mate this helped a ton
Thanks, I really appreciate this
我很感谢 Wǒ hěn gǎnxiè
Translates to I am very thankful/Grateful. This term I'm less familiar with but to me it would be to a way of expressing a large amount of gratitude in a formal manner. Same sort of situations 您 Nín is used over 你 Nǐ
Thank you sir, I appreciate this.
不客气 Bù kèqì
The best translation for this is No politeness necessary or more commonly You're welcome
不用谢 Bùyòng xiè
It literately means "No need thanks"
Don't worry about it mate
It literally translates to "Nothing Thing" but a more natural translation would be it's nothing / never mind
It was nothing mate (As in it was no trouble)
Responding to 谢谢 (xièxiè) with 谢谢你 (xièxiè nǐ) although would be understood it isn't a way I have heard before. I think it would work well if someone who you were in a position to thank thanked you before you said thank you. An example would be purchasing something from someone as you're both equally profiting from the interaction but if the thanks are one sided I wouldn't use 谢谢你
I mean if you helped a lady up after she tripped and then thanked her for letting you help her she would scream 变态 (biàntài) and slap you across the face assuming you took some sick joy in it
Bonus way of saying thanks
note: this is not about the Chinese language but the Chinese culture
When someone pours you a cup of tea rather then vocalizing your thanks you can tap 2 fingers on the table.
This custom is said to have originated in the Qing Dynasty when Emperor Qian Long would travel in disguise through the empire. Servants were told not to reveal their master's identity. One day in a restaurant, the emperor, after pouring himself a cup of tea, filled a servant's cup as well. To that servant it was a huge honour to have the emperor pour him a cup of tea. Out of reflex he wanted to kneel and express his thanks. He could not kneel and kowtow to the emperor since that would reveal the emperor's identity so he bent his fingers on the table to express his gratitude and respect to the emperor.
-Partial Extract from Wikipedia