I'm planning to go to the 2014 World Cup in Brazil. What's cheapest and safest to go between cities?

When you are traveling inter-state the buses usually have better amenities than most cheaper airlines, water/tv/ac/etc. And yes, they are safe to go anywhere. I have traveled from Pernambuco in the Northeast to Bahia and Rio Grande do Norte multiple times, including travels inside of the Pernambuco state.

They could look something like this: http://urbana-pe.com.br/sobre/as-empresas/borborema

It is good to ask in the bus station which services they provide.

Having said that, the country is gigantic, and unless you have a very specific reason not to take the airplanes I do recommend flying.

A cheaper way to fly would be the Brazil AirPass.

If you take the buses in the cities' bus stations, you probably will not have problems concerning the quality of them. As for the safety regarding thieves and speeding buses, thefts sometimes occurs in interstate buses but are not really common. Speeding drives on the other hand are quite common. Also, if you do decide to go by bus, consider that the distance between Belo Horizonte and Curitiba is much bigger than to Sao Paulo, which is still quite distant. You best option could be taking night buses from BH to Sao Paulo (around 8-9h), then go from Sao Paulo to Curibiba (around 6-7h). If possible, don't leave on a friday night to avoid traffic.

Otherwise flying is much faster and if you buy in advance not much more expensive, although you should also consider the costs of going to and from the airports, which are usually farther from the city's centres.

For other touristy cities that don't have airports, there are usually good to ok buses to get there. For some small towns you might have trouble going if you don't rent a car, but it's very unlikely you'll go to places like these.

I've never been to Brazil, but I will be there for the World Cup. I have found http://www.buscaonibus.com.br/ to be a pretty good resource to find bus routes and booking fares on the major bus companies.

Flying right now, as of March 2014, is getting pretty expensive (even on domestic/low-cost airlines) and since the cheaper seats have been taken. Unless you have lots of reales to spend on airfare, or you already have flights scheduled I would recommend booking bus fares ASAP. You kinda have to balance the expense of the flight, versus how far away your destination is, and compare that to the bus fare and time on bus to make a decision as to what's best for you. Some people are happy to spend 12+ hours on a bus. Not me!

Update: BTW If you're over 21, you could also rent a car they usually run for about USD$50-60 per day (for non-youthful driver, ie 25+). You need to be a good/responsible driver, and prepared to pay for gas, and possibly drive a standard transmission car, and deal with traffic.