Hope this isn't too confusing. The key switch is a generic 5 pole for lawn and garden equipment. DO NOT* use an automotive switch. If you use the Briggs version the terminals are different, most shops stock the generic so I've used it here.
The "M" is for motor and is used to ground the ignition to shut off the engine.
"A" is for accessories such as an electric clutch and you're not going to be using this.
"R" is for regulator or charger and as of now this won't be used.
"S" is for solenoid and will activate the starter solenoid.
"B" is for battery and will always be hot, recieving battery voltage from the in side of the starter solenoid.
*If you use an automotive switch it can direct battery voltage to the ignition coil and will destroy the coil in an instant!

I've shown two starter solenoids, one a four pole and one a three pole. These are also rather generic but also refered to as Ford type. Either will work fine. When wiring the unit just block out the version you aren't using. The two poles at the end are for the larger battery cable sized fittings, the one or two towards the center are for smaller wires like the key switch.

The key switch terminals are usually slip fit but I've seen screw type. The solenoid and starter motor will be post and nut. The kill wire terminal at the throttle bracket may have a post and nut, slip fit or spring loaded clamp.

You'll need to do a bit of homework with figuring cable and wire lenght needed, terminal type connectors and if you need to get a crimp tool for the terminals. I'd advise getting the components first so you'll know exactly what you'll need for hardware before purchasing those items. If your dash where you will mount the key switch is not grounded to frame, find a key switch with a 6th terminal mark "G" for ground so you can run a ground wire to frame. Actually you could use the 6 terminal switch even if you don't need the extra ground so if that's what's available at the shop you go to, use it.

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TVS/TNT Linkage
This top drawing is not of your engine but the set up is very similar. Dis-regard the choke plate in the first drawing, it is for a TVM model. The details are much clearer than the picture of your engine which is the second drawing. Both show the throttle bellcrank at full throttle, if the bellcrank was rotated clockwise to idle, the hole the link would connect to would be closest to the block and towards the spark plug end of the engine.

If you should have loosened the governor arm you will need to re-set it. If you haven't, don't. To set it loosen the 1/4" screw, rotate the bellcrank counterclockwise, the governor arm to the left, and the governor output shaft counterclockwise. Hold them all securely in that position and tighten the 1/4" screw. The screw needs to be tight but use caution not to overdo it, it doesn't have a nut backing it up, just a tin clamp and it will strip out easily if tightened too much.

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SPRINT/CLASSIC CARB LINKAGE

The small spring is not used on earlier models. It was added in "97" as a governed idle.

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25 CUBE VERTICAL LINKAGE

Top link is throttle, bottom is choke. The choke slides into a slot in the control plate.

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13 VERTICAL CUBE LINKAGE