First of all, to acquire permanent residency in Paraguay you need to deposit 25 million Guarani with the central bank (around 4400$US). You need a few other documents such as a criminal background check, birth certificate etc.
You need to fly there and apply in person (on your own or with the help of a local immigration attorney). You can complete the whole process in a few days but I recommend spending a month in town just to make sure.
A few months later you receive your ID card (cedula) and voila, you are a PR.
That being said, you can’t become a citizen easily. The only way at the moment is to spend three years IN the country and prove significant local ties (a local business, a wife/husband, lots of friends etc).
I wrote an article on the subject a few months ago: