How to say thank you in Chinese:
谢谢你 xiè xiè nǐ

How to say sorry in Chinese:
对不起 duì bù qǐ

Other related terms:
不用谢 bù yòng xiè - used as a reply to "Thank you"; literally means "no need to say thanks"; similar to "No problem".

不客气 bù kè qì - used as a reply to "Thank you"; literally means "no need to be polite"; similar to "You're welcome".

不好意思 bù hǎo yì sī - term that you use when you made a faux pas or you are doing something that may inconvenience others; similar to saying "mea culpa" or "excuse me".

没关系 méi guān xì - used as a reply to "Sorry"; literally means "it's not a concern"; similar to saying "It's alright".