As freelanceastro noted in his answer: Yes, you can measure the cosmological redshift of quasars with simple equipment.
The page he referred to is not a fabrication. (See the article on amateur astronomical spectroscopy in the August 2011 issue of Sky & Telescope.) Many amateurs have made this measurement with a variety of instruments, including a modified security camera on an 8" SCT. (That one took just 15 minutes total integration time, which is about what most most amateurs find using typical equipment.)
As you noted, asawyer, quasars are good targets for red shift measurements because their radial velocity is so high. The Doppler shift of objects that are closer than 3C 273 is more difficult to measure because their radial velocity is so much lower.
As astromax notes, capturing the spectra of extended objects generally requires a slit, but not always! For example, a lot of amateurs captured spectra of comet ISON, which certainly isn't a compact object! Check out this one, done with just a DSLR and an 80 mm refractor:
If you're interested in 3C 273, in case you missed it, there's a wonderful transcript of an interview with its discoverer, Maarten Schmidt here: http://www.aip.org/history/ohilist/4861.html. It's rather long. Search the text for the text "December 1962" for the relevant section. It's a fun read about how he made the discovery and his personal experience of the process.