I don't really consider this a short trip, not compared to the flight. The only real ground transport option in South America is the bus (coach). There are some domestic trains around the big cities on the eastern coast, but for most rides you have to take a bus.
I took a bus from Salta in Northern Argentina to Asunción and that took about 24 hours (including some bus changes).
I don't think there is a direct bus from Santiago to Asunción, most likely you have to change in Mendoza and/or Cordoba.
There are several different bus stations in Santiago, make sure you go to the right one. All the big ones are close together next to the train station.
Also my bus took me as far as the Argentinian border town with Paraguay 'Clorinda'. From therr you can take a cab, or like me walk to the border and take a local bus to Asunción which took another few hours.
If you do that, don't go into town and try to use the foot bridge over the river into Paraguay, that crossing is for locals only. You have to go through the big border crossing out of town.
Anyways, remember you are crossing the Andes on the way which is spectacular. Some of the landscape in Northern Argentina is also nice, but if you go via Cordoba you may miss some of it.
I would say the ride takes at least two days, but you should stop in Mendoza which is really nice.