I've created an In Design document with two columns. However, do I have to draw separate text boxes in each column to get the columns to work?
You don't "have" to, but that is one way to do it. You could draw one big text box and 'split' it into 2 or more columns. This would then allow you to span columns without drawing additional boxes and to also use the column break to move content from one to the next. Neither way is wrong, just depends how you work and what is easier for your layout.
If you drew different text boxes, one for each column, you would simply link them together to get the text to flow from one to the next.
Learn how to use masterpage textframes and how to plaxe text, or learn the alternative way to place text with Shift or Alt-Shift.