Invercargill is a town on the very south of New Zealand's South Island. It's main industries are meat and aluminium. There's not a lot to do in Invercargill but many of surrounding areas are quite beautiful.
Hitching is easy enough from Invercargill.
Heading West and Through Central Otago
Heading North towards Central Otago (Queenstown, Wanaka, etc) or the West Coast (Greymouth, Westport etc.) just take State Highway 6. State Highway 6 is the main street through the city, in the south of the city it's called Clyde Street, in the middle of the city it's known as Dee Street and at the north end it's called North Road. The road is quite wide in most places and there are many places for cars to see you and space to pull over.
Heading North and Via the East Coast
State Highway 1 goes through most of the major cities in NZ. Hitching north to Gore, Dunedin, Timaru, Christchurch this is the way to go. It's usually fairly easy to get a ride when you wait a hundred meters or so down from the corner of Racecourse and East Roads (East Road is part of State Highway 1).