Its not very practical to generate mains voltages with a vehicle alty, better to use it as it was intended with a 12v battery, and use car 12v bulbs for your light and maybe a car charger for your 'phone.
The case of the alty requires a ground (battery -), the heavier terminal goes to battery +, and the snall terminal goes through a 1.2w 12v bulb to battery + via a switch to start it generating.
This is true for an older 2 wire alty, newer digital 3 wire ones are a lot more complex and require special signals to make power and do not have regulation built in.
Some care needs to be taken here as an alty can make a lot of power, a job for someone with experience.
A weed eater might be too snall, you need around 3hp for a alty, might work if you gear it down with your pulley arrangement.