Asunción is the capital city of Paraguay.
North towards Filadelfia
Get on a #5 bus that heads out towards Villa Hayes. It is likely that you will have to take a bus from the centre to Ruta Transchaco to get this, for example Linea 24 or just ask about. #5 goes out to Villa Hayes. Alternatively, ask for any buses que crucen la puente, the only bridge over to the Chaco. You can hitchhike from the roundabout 1km after the bridge, or stay on the #5 and it will keep going along the route. At Benjamin Aceval (some 5's go as far as here, ask), there is a peaje, and the police is cool about you hitchhiking here. If you get here late, you can camp safely nearby.
From Ciudad del Este
Walk to the west, away from the crowd that enters the city to go shopping. Most of the people who enter the city go back to Brasil at the end of the day. There's only one road to Asuncion, so have a little patience to meet somebody traveling to the capital. With a bit of luck you meet one of the so many friendly people in this country, who will love to talk about their country.
Proceed as described above, turning south at the roundabout after the brige. Alternatively, on Ruta Transchaco, get a bus towards Puerto Falcón (it should say "Falcón" as a destination), it costs PYG 5000 (03/2018), walk across the border and hitch a ride towards Formosa (it seems that RN81 in Argentina gets way more traffic than RN86 directly from Clorinda).
South towards Encarnacion
Pick up bus #18, which runs close to the bus station and from various points, or bus #85, which stops right in front of Paseo La Galería shopping center on Avenida Santa Teresa, as it heads south. Guarambare is what you should ask for, or whether they go along Acesso Sur. It is a 40-50 minute run out of the city. Get out if you see a good spot, or if the bus pulls off the main highway. Trucks use this route into the city and there are a lot of warehouses.
Places to avoid
A fairly safe city on the whole, even though locals indulge in a bit of fear mongering. Central area is generally very safe in the night. Leave plenty of time to get out of the city fringes house a lot of poor communities.
Accommodation and Sleep
There are many plazas in the center. As of July 2014, large flooding has forced many barrios to relocate into the center. They are safe and friendly people on the whole. Near the train station there was a tent village.
Other useful info
Supermarkets are very good value here, so stock up before you head out into the stix. Free internet (computers), maps and toilets in a good tourist information place in the center.
It doesn't hold too much appeal for the traveler.