It's not clear to me in Lesson 13 whether I should begin the exercises in chest tone or head tone. When I begin in chest, I break through into head and my voice will crack. If I start in head and stay there (singing falsetto) it's easier. But I do wish to expand my chest tone range.

In general, attempt to sing your lowest pitches in chest voice, and your highest in head voice! See this tutorial for some guidance on developing mix (for the transition & middle pitches): https://www.30daysinger.com/tutorial/mixed-voice-101/1. 

It may also help to sing on a syllable with a consonant (like DOO or GEE, rather than just a vowel). Also consider singing on something semi-closed (lip trill, NG, etc). Both of those modifications can help you achieve balance and minimize cracking; but cracking is very normal and to be expected, before mix is developed!