"Money isn't important to me" is a very natural sentence.
"Money isn't a thing to me" may be appropriate in casual conversation.
"I don't appreciate money" is a reasonable English sentence but it means something different. "Money isn't important to me" is a way of stating that there are other things more important to you. It's a positive statement about your values. "I don't appreciate money" sounds like a negative statement about yourself, as though you ought to appreciate money. Similarly with "I don't value money." It sounds like a negative statement about yourself. (It's reasonable to value or appreciate money to some extent even if we consider it not the most important thing in our lives.)