Transport in South America is fairly extensive and easy to use, whether your backpacking tour is taking you far afield or if you're only travelling locally. There's good tourist infrastructure in place, so it's usually easy to reach even the most remote areas (though you might pay more for the privilege).
If you're sticking to a strict itinerary it's often a good idea to use domestic or other short distance flights between destinations, as overland travel in South America can be time consuming. However flights between countries in South America can be expensive compared to domestic flights. This means it's sometimes cheaper to fly to the border city of one country, cross the border on the ground, and then take another flight to your final destination.
Make sure to check out visa requirements before travelling.
It's usually easy to get reasonably cheap flights between major cities within a country. Exercise caution when it comes to using really low-cost regional airlines, as some of these have less than stellar safety records.
Railways in South America are not as extensive as elsewhere in the world. In fact there are no cross-country train services at all on the continent.
The more developed countries like Argentina, Chile, and Brazil have reasonable domestic networks that generally connect regional capitals, and these can be a good option if you're travelling on a budget. Other countries like Bolivia only have trains on very specific routes, some of which are scenic routes designed just for travellers.
If you're looking to travel overland in South America, using buses is definitely your best option. There are numerous bus companies that will take you all over the continent, and usually for a very reasonable price. Of course these journeys can take a long time, so for longer distances you should consider flying.
The quality of roads in South America varies enormously. In more developed countries there are good road networks, whereas in countries like Bolivia and Guyana you'll often find roads in disrepair, or get stuck with unpaved tracks. We generally don't recommend driving in South America if you can help it.
Taxi drivers in South American will often try and overcharge you. To avoid this it can be worth asking around to get an idea of what you should be paying for a given journey. Unfortunately there aren't really regulations about meters like you often find elsewhere. Agreeing a price before travelling is often a wise move.
As for safety, in bigger cities taxis can actually be a safer option late at night than taking the bus or walking. If you're unsure about the safety of taxis in your area, visit the local tourist office.