I have a project built up and working. My next step is to make it more 'permanent'. What is the recommended way of making permanent connections? Seems like just inserting wires into the holes in the headers isn't real permanent.
Veroboard ( stripboard) or custom design printed circuit board with components soldered in either through hole or surface mount or make your own.
Premade modules with header pins ( soldered connection for more permanent if you wish.)
There are mechanical AND electrical considerations.
Here's a look at mid-level 'permanent' approaches:
That is mostly wood, attached to a bookcase.
But there ARE times for HEavy Metal:
See: Project-GlassKilnController - ArduinoInfo
That project ended up mounted in a large metal box along with power control (Thousands of watts of Kiln Heaters).