Hi Claire,
There is no need to join a tour to travel from Rio to Buenos Aires via Iguazu or from Buenos Aires to Bolivia. The bus ride from Rio to Iguazu takes about 23 hours. The bus ride from Iguazu to Buenos Aires 18 hours. You should be able to make it to Buenos Aires in 3 to 4 days.
In Uyuni tours are widely available to visit the breathtaking salt flats. Do try to go to Rurrenabaque for a jungle trip while you're in Bolivia. It's the best place to see an abundance of wildlife. Getting there is quite hectic, you'll be traveling on the infamous death road, so do ask about road conditions before going. Also have a look at
Potosi in Bolivia, a visit to the silver mines is highly recommended.
In Peru you can travel from Puno (Lake Titicaca) to
Arequipa for the beautiful Santa Catalina monastery and Condor sightings before going to Cuzco. It's a long way from Cuzco to Lima so you may consider taking a flight to avoid spending days in a bus.
You'll have lots of fun in South America. Keep us posted!