Solution 1:

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:

For example:

  • 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:

For example:

  • 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).

Solution 2:

It's a bit tricky but you have to learn to tell whether the "To" is being used as (A.) part of an Infinitive or (B.) a Preposition.

Once you develop this skill, you can follow this:

Infinitive "To" = to + base verb

Preposition "To" = to + Noun, Pronoun, V+ing

Examples of Infinitive "To" are:

would like to + base verb

plan to + base verb

decide to + base verb

etc.

Examples of Preposition "To" are:

go to + Noun

look forward to + Noun/ V+ing

be used to + Noun/ V+ing

allergic to + Noun

etc.

Hope this helped!

;-)