The simple way to set up an HP DeskJet 882C with Linux is described here.
It uses the standard HP DeskJet 550 device driver and gives decent results.
To get slightly better resoution out of your printer (you can see the
difference by comparing the results side by side), you need to
apply some patches to ghostview to get a device driver specificially for
the HP DeskJet 880 series.

Use printtool to set up the printer:

as root, run "printtool", select "add", select "local printer".  Used the
default printer name of "lp" (using hardware printer device /dev/lp1).
Then click to "select" the input filter.  Selected "HP DeskJet
550C/560C/6xxC series", with 300x300 resolution, Color Depth 32 for
"Floyd-Steinberg Color Printing" (looks better in the color postscript
tests than color depth 3, and doesn't seem to be noticeably slower than
color depth24).  Selected the printing options "Send EOF after job to eject
page", "Fix stair-stepping text" (does LF->CR/LF), and "Fast text
printing".

Print a test postscript page in printtool, or do it manually with:

lpr /usr/lib/rhs/rhs-printfilters/testpage.ps