What are some tips for cheaply but tastefully furnishing one’s first apartment?

Shop around at places that sell used furniture. Check out estate sales. Don't worry about finding matching pieces.

If the cost is too high, make a counteroffer at a price you can afford. Take your time and find things you really like.

Avoid used mattresses. There is too high a risk of sanitation issues or importing hidden bugs into your home. Many states have laws forbidding the resale of used mattresses unless they have first been sterilized.

Used sofas are probably not a good idea for most people unless you are prepared to completely reupholster it.

There are plenty of inexpensive new sofas that should be fine for a few years until you are ready for something better.