Thrift stores, basically. I worked in various thrift stores for 5 years and mostly furnished my home that way. Just be thorough when you're inspecting it before you buy it! Check for bugs, broken/missing pieces, wiggly legs, cosmetic damage. We've gotten some really cool stuff in great condition for actually reasonable prices. My favorite is an ancient La-Z-Boy for $12, in decent shape.
If there's a Re-Store near you, they're great too! We got an adorable coffee hutch for $65, a decent couch for $80, a couple nightstands for $10-20 each, a solid, heavy duty coffee table for $30. It requires patience to find what you might be looking for, and it helps if you already know which thrift stores tend to get better quality items. Antique stores are worth a shot too, if you're ok with spending on the upper end of cheap? You can get some really well built, old, beautiful furniture. Or see if there's a furniture consignment store around you!
I think things like IKEA or Target are fine too, as long as you take care of your furniture. I've never had any problems with anything from IKEA, even when I buy used, but it probably helps I don't have kids, lol.
Same here! Our whole kitchen had at least 2 sets of owners worth of grease and grime coating everything. The people before us only owned the house for a couple years, but I don't think they cleaned the place even once. We spent the first day of homeownership scrubbing all the nooks and crannies of just the kitchen. The outsides and tops of the cabinets are still a bit sticky, but we're hoping to replace them soon. One of the burners on the stove was absolutely encrusted with burned stuff, the bottom of the fridge drawer had tons of dust and crud, and the walls were covered in handprints, food splatters, and misc. grime. Bleh!