When should "into" be used rather than "in to," and vice versa?

You should use "into" when it's a question of location, for lack of a better word. "I went into the store," "We went into the field of computer science," "We drank well into the morning," etc..

"In to" just happens sometimes. "I went in to buy some milk." In that sentence the "to" is part of the infinitive "to buy."

If you aren't sure which one to use, change the "in" to "in order" and see if it still works. "I went in order to the store" is wrong, but "I went in order to buy some milk" is good.

This is explained in the book called "Common Errors in English Usage" by Paul Brians:

“Into” is a preposition which often answers the question, “where?” For example,

“Tom and Becky had gone far into the cave before they realized they were lost.”

Sometimes the “where” is metaphorical, as in,

“He went into the army”

or

“She went into business.”

It can also refer by analogy to time:

“The snow lingered on the ground well into April.”

In old-fashioned math talk, it could be used to refer to division:

“Two into six is three.”

In other instances where the words “in” and “to” just happen to find themselves neighbors, they must remain separate words. For instance,

“Rachel dived back in to rescue the struggling boy.”

Here “to” belongs with “rescue” and means “in order to,” not “where.” (If the phrase had been “dived back into the water,” “into” would be required.)

Try speaking the sentence concerned aloud, pausing distinctly between “in” and “to.” If the result sounds wrong, you probably need “into.”

Then there is the 60s colloquialism which lingers on in which “into” means “deeply interested or involved in”:

“Kevin is into baseball cards.”

This is derived from usages like “the committee is looking into the fund-raising scandal.” The abbreviated form is not acceptable formal English, but is quite common in informal communications.

Broadly speaking, in refers to something that already exists inside something, while into implies motion from outside to inside.

Dexter was in the room at the time of the murder.

Dexter went into the room shortly before the murder.

People often use in instead of into, especially if in is preceded by an adverb:

Max went down in the mine with the rest of the crew.

This is an informal usage, but you will hear it a lot. Nevertheless, if you follow the general rule listed in my first sentence, you should be able to understand the difference and make yourself understood.