I just went through this with my 56. The biggest problem will depend on whether or not you have a 57 engine in the car. The 57 engine block has no provisions for bolting up a starter (it is on the bell housing). The TH 350 has no way to mount a starter.
I ended up using the starter adapter mounting plate sold by Classic Chevy International (and others I am sure). It is a 1/4 thick steel plate that fits between the engine and transmission.
You will also need a transmission mount adapter plate. Something similar to the adapter plate that was used on 1962 Corvettes, but remember, your transmission is pushed back 1/4 inch so you will need to slot the holes 1/4 inch.
Also, you will need to find a way to mount a shifter. The TH350 does not have mounting holes for a floor mounted shifter. You will need to make some sort of bracket to mount the shifter.
I made a combined transmission mount adapter plate and a shifter mounting plate. It worked out well.
Also, the TH350 will need an external cooler, so you will need to run cooling lines to the radiator area. I used an aftermarket trans cooler.
And last, you will need to get a vacuum source for the modulator valve. I put a fitting in the base of the single four barrel carb.
Not a exceptionally difficult job, but quite a few details to think about.
FYI: Your TH350 must be the short tail housing version, and you must also change the driveshaft yoke.