Looking for an idiot's guide to texture baking (dissimilar geometry)

I’ve got a old character model where I’ve redone the geometry on a standard base mesh (as the inter-compatibility will be very useful for what I’m doing); This was mostly done using a shrinkmap modifier and carefully shifting around vertices on the new mesh to best match the contours and features on the old shape.

I now need to bake the textures across to get the new model to match up as closely as possible to the old one.

However, whenever I try to do this, I’m getting completely blank textures or complaints about circular references in the texture stack (implying that the model is trying to copy textures from itself and that I’m somehow not correctly selecting/setting the master that I want to copy the textures from).

So, I went looking for guides to check what I’d done wrong, but these are mostly buried under guides on baking to a new UV on the same geometry, and of the guides I’ve found that are on the right subject, most of them are silent Youtube videos where the uploader doesn’t actually explain what they’re doing.

Can anyone please point me at a guide that will actually help me work out what I’m doing wrong?

TL;DR: I need to bake textures between two meshes with very similar shape, but different geometry.

EDIT: I’ll add, I’m still on Blender 2.79b. Not enough of the add-ons I use have been updated to the 2.8 Beta yet.

What you’re looking for is a simple emission BSDF, bake selected to active type emission. Do not connect your new texture to anything while you bake. Save yourself some time and set the render samples to 1. You don’t need more for emission bakes.