In English, the form V + -ing is called a gerund if it serves as a noun. For example, the gerund form of "run" is "running". (I like cats, I like dogs, I like running). However, not all V + ing forms are gerunds--in "I am running", "running" is another verb.
This blog post explains that there are a few cases where you use to + V + -ing:
1) If the to is part of a phrasal verb or verb + preposition combination:
A phrasal verb is something like "look forward to", "confess to", etc. In this case, the "to" is part of the verb itself. Some examples are:
- I am looking forward to attending the party
- I am committed to supporting the Linguistics proposal on Area51
2) If the to is part of an adjective + preposition combination:
- A lot of women aren’t accustomed to being treated so well!
- I am opposed to increasing taxes.
These look similar to the examples above, but you can tell that a word like accustomed is an adjective in this case because there is a form of the verb to be. That is, you are an adjective.
3) If the to is part of a noun + preposition combination:
- His addiction to gambling has caused a lot of stress for his family.
- Her great dedication to teaching inspires her students.
Going through your examples, "I would like to apply what I learned in school to helping make more money" is incorrect, and doesn't fit into any of these categories (hat tip: Karl Knechtel).