Dunedin to Invergargill
Rather than cycle out and criss-crossing the motorway heading south and down the main highway, I suggest you head out through the suburbs of Dunedin and follow the coast road south. This is gravel and rough in parts but often very scenic and without heavy traffic.
Taieri Mouth (about 40 km from Dunedin, campground) is a beautiful spot with a great coastline. Carry on along the Milton Taieri Mouth road (another 30 km or so) and camp in Milton. The following day, cycle out along the Milton-Toko Mouth Road to Toko Mouth (about 20 km), another lovely spot, and then head along the Wangaloa-Toko Mouth road to Kaitangata (another 25 km or so). This is a sweet old New Zealand town with the remains of the old railway line and ornate villas on large sections. The store has great ice creams but alas, no accommodation. Follow the sealed road inland (another 25 km) through Stirling and camp at Balclutha (supermarket, hostel, campground).
Overall, this is a taxing trail in parts, not for the hills, but for the stretches of gravel roads. However, in my opinion, it beats heading down the main highway and introduces you to some of the lesser known and very lovely parts of this country.
The Caitlins is one of New Zealand's great beauty spots. There are a number of lovely beaches to explore and this trip covers just a few of them. A few hills rise sharply to 200 m but the road is generally rolling and not too strenuous. The road geography of the Caitlins allows many options. Generally the sealed main road winds its way just inland from the coast. There are a number of roads (often gravel) leading out from the main road to the beaches and campgrounds, but mostly you have to come back the same way.
One option I suggest is that you take the road to Kaka Point and spend the night there (about 30 km campsite, pub and store). The same day you can take a trip out to Nugget Point and the lighthouse and back (about 18 km). Then carry on through Owaka (about 15 km) and follow Highway 38 to Papatowai (25 km, campsite and store). Another 50 or so kilometres will take you through Waikawa to Curio Bay and this is a gem of a place. The campground, albeit a little primitive, has marvellous views, but the great thing is that you can swim with the dolphins on one side of the hill, or walk over to the other side and explore the petrified remains of prehistoric forests in the foreshore. The following day, head out along mostly flat roads through Haldane and Fortrose (about 30 km, no stores) and then Invercargill (a further 45 km).
This is only one option. I suggest you get a map of the Caitlins in Balclutha and check out other great side trips including sights like the Cathedral Caves. There are several impressive waterfalls and a number of DOC campsites, but the shops are few and often far between so take supplies. The coastline in this area is long and empty with a turquoise sea and breakers of intense whiteness breaking onto a broad foreshore of white sand.