Before you try to enter Paraguay, you need to check the visa requirements for your country. Most European citizens EU generally don't need visas to visit Paraguay for up to 90 days. However US, Canadian, Australian and New Zealand citizens do need visas.
Citizens of the US, Canada, Australia, New Zealand, and Taiwan can apply for a visa on arrival, but only at the International Airport in Ascunción. Fees range from USD 100 to USD 160 in cash, depending on nationality, and the maximum stay on such a visa is 90 days.
Bus service is available to and from a wide range of South American cities. You can take a bus from Santiago, Chile; Sao Paulo, Brazil; Buenos Aires, Argentina; Cordoba, Argentina; Santa Cruz, Bolivia; Montevideo, Uruguay; Rio de Janeiro, Brazil. Buses are very modern and some buses have seats that fully recline into beds. In Spanish they are called cama. Semi-cama recline most of the way.
The bus to and from Bolivia crosses the Chaco. Only one bus company does the trip, from Santa Cruz to Asuncsion, with possible stops at Villamontes Bolivia or a few kilometers of Filadelfia Paraguay but typically for the same price. Beware that tourists should bargain, with minimum prices of 45 US dollars payable in local currency or in US. The trip takes at least a full day, and easily more. Note that the bus comming from Santa Cruz passes Villamontes at about 2 in the morning.
The bus from Ciudad Del Este Paraguay at the border with Brazil is typically cheaper for going to Rio or Sao Paolo than the bus going from Iguasu Brazil. To go to Puerto Iguasu Argentina many busses cross Brazil without Brazilian border control into Argentina for only 10 Pesos or 10.000 Guarani. Best visit the border in advance to get an exit stamp for Paraguay because not all busses make a stop, otherwise ask before jumping on a bus.
Flights go out from other South American airports to Asuncion on a fairly regular basis. There are also two daily flights from São Paulo to Ciudad del Este.Direct flights between Miami and Asunción are currently available on a regular basis. Direct flights between Madrid and Asunción are scheduled to start on the first semester of 2014.
A boat travels up the Paraguay River from Asuncion stopping at many ports to the north of the border with Brazil. Most of these boats weren't designed to carry passengers so expect a sticky crowded experience, but you can buy just about anything on board, even cold beer. Bring something comfortable to sleep on!