Use of "on to" versus "onto"

Solution 1:

Generally, when you are using the preposition "in" or "on" with "to" there is a rule of thumb to follow:

You use into [or onto] to say where someone or something goes, or where something is put.
Abused, Confused, & Misused Words

We can break this down a bit further.

1. If "to" is followed by a noun location (e.g., table, box, room) you can combine the prepositions: onto the table, into a box, into the room. Example:

I placed the book onto the table.

2. If "to" is part of an infinitive verb, use the split form. Example:

Arriving at my sister's room, I went in to find her weeping.

but be careful if you change the word order:

I went into my sister's room to find her weeping.

addendum

A commenter below believes "Onto victory!" is an exception to what I say here, but I will only note that an NGram search shows that to be a seldom-used construction, probably a solecism.

The sense of "onto victory" would seem to me to be not "moving toward victory" but "placing something on top of victory" which makes little sense (although one could argue for a metaphorical interpretation here, that is pretty strained and certainly not the most immediate and familiar potential usage).

Solution 2:

1. "Thank you for holding on to me when I did not want to hold on to anyone."
   
2. "Thank you for holding onto me when I did not want to hold onto anyone."

They are slightly different in that we could legitimately set part of the first one off with commas, or even parentheses i.e.

"Thank you for holding on (to me) when I did not want to hold on to anyone."

This is possible because there are two distinct verbs

(1) to hold

(2) to hold on

This splitting is not possible with all verbs though because not all verbs have a prepositional form.

Example

(a) He stepped onto the platform. (correct)

(b) He stepped on, to the platform. (incorrect)

The second example is wrong because there is no prepositional verb "to step on" . You can use "step" with "on" but not as in sentence (b). For example , it is possible to say, "He stepped on the cockroach". (correct)