If mechanics forgot to put in coolant and it burned up the engine in ten miles, which parts do I need to replace?

All you can do is replace the coolant, and hope it hasn't damaged the engine. An engine that overheats badly can seize. Typically before that happens the engine would begin knocking and the engine's computer would shut if down. If engine was damaged it may need a rebuild or be replaced. Have you talked to the shop that did the work? Are they willing to make it right? You may need to look into a small claims court if their actions damaged your engine.