The two idlers on the rear derailleur are called "jockey wheels." Routing is as you see in VegasTriker's picture - around the left side of the lower one and around the right side of the upper one, then around the rear of the gears at the wheel. The gears are collectively called the "cassette" because they are a stack of gears, which individually are usually referred to as "cogs." To the front, in your case front AND mid-drive, the gears mounted on a spider are called "chainrings."

I'm not going to call the shop incompetent or accuse them of being unwilling to work on a 'bent, for the reason that they might be right. Measure your chains with a ruler. Twelve full links should measure exactly 12 inches. If you broke the chain, chances are that both of them are worn way beyond the point where your cassette and all your chainrings have been ruined. Replacing a chain when your gears are worn will just result in poor shifting, skipping, and quickly wearing out the new chain. Changing them all is something even a noob can do, with a few tools and a little help from youtube videos.

It's time to educate yourself.