What to bring?

While it may be possible to find apartments that are "furnished", unless your budget is huge and your planning on renting some ambassadors former home, your not going to be very happy with the amount/quality of these furnishings.   So yes, I'd budget for basic furniture, which will be significantly less than what you'd have to budget for in the US as long as you buy simple and basic stuff.  As your only going to be here a year or two I wouldn't recommend buying expensive custom or imported furniture as the resale market is pretty sparse.   Occasionally an expat here will advertise on the FB expat site (Expats in Paraguay) and there is usually a few expat "garage sales" each year where you can pick stuff up.

Your on the right track to bring clothing and shoes.   You can get everything here that you can buy in the US (more or less) but you'll pay a lot more for it here and the selection and availability will be limited.  And simple stuff like underwear, socks, etc…are low quality here so stock up on these especially if you have kids.   

Don't worry about kitchenware, you can pick up silverware, flatware, glassware, cookware, all the plastic bins/containers/etc youll ever need 'll day long at any grocery store, and at generally cheaper prices than in the US.   

As for electronics, unless your wanting to bring down your high end stuff it would probably cost more to ship something down from the US than to just buy new here.  And keep in mind the voltages are different here (210/220 Volts as opposed to the US 110 Volts) so unless what you want to bring down is already dual voltage you'll have to deal with hooking it up to a voltage converter.    For costs of electronics here, a mid size flat screen TV's (not talking Sony's) are around $400,  computers are twice the cost as in the US (to include peripherals like printers), cheap DVD players run about $100,  "kitchen electronics" like microwaves, blenders and mixers are about the same cost as in the US.   One thing to buy and bring with you though without a doubt are good quality surge protectors/power strips.  Go online (amazon.com) and order ones that are 210/220 Volt, they are almost all made in China and will run you about $25-$40.   Best investment you'll make as there are constant power outages, power surges and/or fluctuations that will just eat your electronics alive.    The ones you can find here are either of dubious quality from Brazil or really low end Chinese junk, the better ones will be much more expensive than on amazon.com (if you can find them).

Also keep in mind if you bring something here to Paraguay that is "under warranty" you'll have to forget getting any warranty support down here for it.   That's about all I can think of at this point, holler if you have questions regarding specific items.

Thank you, this was super helpful! What about the big appliances? One thing I've read is that a lot of apartments don't come furnished with a stove. Is that the case? Also, is a fridge standard in a rental?

One other question I thought of. Is there an English-language bookstore in Asuncion?

There is a bookstore at the mall "Shopping del Sol" which has an english language section.  We can pick up my son's favorite kids books here and a smaller selection of big people paperbacks, but it's pretty sparse.   There may be other stores too, we just haven't found them as we don't live in Asuncion, only come there occasionally to visit he "big city".    I'd really suggest trying that FB site "Expats in Paraguay".  There's a ton of people on there who are very responsive and most of them live in Asuncion (wimps! %^).

Many rentals will not include the appliances, although you may get lucky.  Keep in mind that stoves are mostly propane (fed through an RV sized canister you swap out yourself) and much smaller than a US stove.  You can pick a new stove up for $400 - $500 and up.   Fridges the same deal, mostly they don't come with the apartment but you can buy new for a little more than a stove.   But they are smaller here than in the US, unless you want to spend big bucks.  Same thing with a washer and dryer, they are like "mini apartment" sized here for the most part.  After living with the ones we bought here for like a year we bit the bullet and ordered US sized ones that were 220 volt and shipped them.  Pricey but worth it.  Now you can actually buy US washers and dryers here at the larger appliance stores though but they run around $500 each last ones I noticed.

Thank you so much again. Your responses help put everything I've read into context. Are a washer/dryer an absolute necessity, or are there "lavanderias" readily available?