You are attempting to describe an action, but you can’t remember whether one, say, goes in to the breach or into the breach, or whether one, for example, walks on to the next trail junction or onto the next trail junction. This post explains the respective roles of the operative words and phrases.
A prepositional phrase is a phrase that includes a preposition, a word that, as its name implies, comes before the object of a sentence. For example, into and onto are prepositions describing movement in relation to objects in the prepositional phrases “into the trench” and “onto the roof.” Because these prepositional phrases provide additional information about an action that occurred (as in the sentences “She leaped into the trench” and “The boy climbed onto the roof”), they serve as adverbs.
We usually think of adverbs as single words (as in “She leaped impetuously” or “The boy climbed quickly”), but adverbs can consist of two or more words, which appear without any additional information or combined with one-word adverbs, either adjacent to each other or separated by the subject and the verb (as in “She leaped impetuously into the trench” or “Quickly, the boy climbed onto the roof”).
“In to” and “on to,” by contrast, each contain two distinct parts of speech: an adverb followed by a preposition. In and on follow a verb to provide additional information about it, and to precedes the object that follows the verb and its adverb, as in “She leaped in to search the trench” and “The boy climbed on to get a better look.”
In the following examples, a preposition is mistakenly employed in place of an adverb-preposition compound consisting of the same letters as the preposition. After each sentence, a discussion, followed by a revision, explains the error.
1. His wife, under the assumption that she’d never see him again, has moved onto another man, and they’re planning on getting married.
The prepositional phrase “moved onto” implies that the woman has literally relocated herself on the man’s body. However, “moved on to” includes the idiomatic phrase “moved on,” meaning “transitioned”: “His wife, under the assumption that she’d never see him again, has moved on to another man, and they’re planning on getting married.”
2. His job is not to give into the demands of multimillionaire celebrities pushing a social agenda.
The use of into implies an entrance, but the key of this sentence is the idiom “give in,” meaning “submit,” so into must be broken up into its constituent parts: “His job is not to give in to the demands of multimillionaire celebrities pushing a social agenda.”
3. The cell phones were turned into the authorities.
Here, into preceded by turned suggests a transformation, rather than the act of turning in, or handing over, so again, in must be separated from to to form part of the phrase “turning in”: “The cell phones were turned in to the authorities.”
For a more detailed discussion about this issue, see this post about prepositions.