1. appreciate后面的确不能加不定式，而一般加名词/代词/动名词/宾语从句。

2. appreciate如果后面加上to sth，这个to也是介词，to sth是介宾短语而不是不定式。

但是appreciate to sth也是没有的，不知道楼主是否能够给出具体链接，也许是你看错了。

3. appreciate一般只要求掌握两种用法：

1）感激/感谢：

appreciate sth：I appreciate your help.

appreciate doing sth：I appreciate getting your help.

appreciate sb/one's doing sth：I appreciate your/you helping me.

appreciate it + 从句：I'll appreciate it if you can give me a hand.

2）欣赏/赏识/重视/鉴赏

You can't appreciate English poetry unless you understand its rhythm.

你若不懂英文诗韵律, 就无法欣赏英文诗。