How do I re-tile a fireplace surround?

One part of your description has informed my response: your mention of "builder-basic."

A basic tile almost always means a basic installation, which usually means mastic (or glue) and not a thin set mortar.
What that means is there is the possibility of movement between the wall/floor and tile which will cause both layers (old and new) to fail. Hate to say it, but I think it's best in the long run for you to remove the old tile all together.  Luckily for you there is a great tool that is available to remove tile and cut grout joint (it has recently become very affordable; I paid $ 45.00 for mine). Look for an "oscillating multi-tool".  They are available from Bosch, Dremel, Harbor Freight (cheap) and the inventor Fein (expensive). This tool and it's pricey blades will make this job go quite smoothly. Once the tile is removed, you should prime the drywall with a product called Guardz ( Zinser) before re-tiling.