Most spider bites are such minor puncture wounds that there may be no bleeding. Read about the treatment of small wounds here and about the treatment of spider bites here.
Non-sensationalistic information on spider bites can be found at the
following linked sites for the five kinds of spiders whose bites can
cause serious problems. (Only two of the five are native to the U.S.)
Widow spiders (especially the two species commonly called "black widows") National Institutes of Health MedlinePlus
Recluse spiders (especially the species most often found in the U.S. and commonly called "brown recluses") National Institutes of Health MedlinePlus
Tarantulas (of which only a few imported species give bites severe enough to require medical treatment) National Institutes of Health
MedlinePlus (for species native to the U.S. only) For sometimes
commercially imported tarantulas giving severe (but not
life-threatening) bites see section 3 of the paper by Escoubas and Rash
at this link.
Australian funnel-web spiders, National Institutes of Health MedlinePlus (I have not been able to find any reports of these spiders stowing away in shipments to other countries.)
Brazilian armed spiders Rick Vetter, University of California at Riverside. (These spiders sometimes show up in shipments of bananas or other produce, as do look-alike species.)