I bought a Briggs & Stratton Vertical OHV Engine with Electric Start 17.5 HP, 1in. X 3 5/32in. Shaft, Model# 31C707-0005-G1 to replace the engine on my Swisher trail mower. The old engine was a pull start so obviously this new engine has more wires. I bought a 5 tab switch and I did get most of it wired.
So here are a few questions I have.

1. I need to buy a starting solenoid What do I need? Size etc?
2. There is what I think is a voltage rectifier the black device in the picture it has a single red wire coming from it. Do I run that back to the battery?http://i1091.photobucket.com/albums/i388/Britwit1960/IMAG0120.jpg?t=1306770901

The single black wire is the kill wire.
The gray wires goes to the fuel solenoid.
The red wire from the regulator goes to the hot side of the starter solenoid.
Any 12 volt solenoid will work.
Luck

Thank you I thought that's where it went but wanted to confirm that. Thank you so much!

What is the proper gauge wire to rum from the battery through the solenoid to the starter?

I would use 16 gauge.
Luck