perhaps the other resondents know something I don'....Either that or they didn't read your inquery close enough....but there's one critical detail.......
YOU CANNOT DO A 3 OR 4 DAY TRIP DOWN GRAND CANYON.
There are shorter trips offered, where you can do the upper section, (lees ferry to phantom ranch (you get to hantom the morning of day 7, and hikde out bright angel trail, 9 miles...river to rim, or you can hike in at hantome, and do an 8 day trip lower slightly more than half,..but I guarantee, if you do one or the other, you'll be back to run the whole thing.
Grand Canyon is a spectacular river trip, and the longer the better. If you only got 3-4 days to do a commercial trip, there are some fantastic river tris you can take in 35 days...there is no 'best trip,' they all have great features, amazing scenery, wildlife, indian ruins and rock art...and Grand Canyon is not the place to DIY, if you have little to no experience with logistics.
If you pm me, I can steer you to some spectacular wilderness and or backcountry river trips...doable in 3-6 days, and outfitters who run them. Grand Canyon is way too special, not to do entire trip (at least to diamond, mile 225). There may be some shorter trips below diamond creek, but 99% of all the great side excursions, all the BIG rapids, Redwall Cavern, Elves Chasm, Havasu, Deer Creek Falls, etc. are all berwqeen lees ferry (mile '0') and diamond creek (mile 225) which is the first lace you can take out...there's no road access between mile 0 and mile 225.
There are so many spectacular river trips...it all depends on your budget (time and $$) time of year, and what you are interested in.
(BTW, Ive run grand canyon as a licensed oar guide, and a private boater where we got a permit, used our own boats and did out own trip...about a dozen times total, and I still have yet to see Grand Canyon from the rim!
I believe if you do a grand canyon motor trip (36ft rafts with 15+ people aboard, you can see whole stretch in 8 days...but why on earth wouldn't you want to spend as much time as possible down there? 4 days. 14 days. 24 days...its still kind of a bummer when you arrive at takeout, on any pristine river trip.
I'm not sure if you wanna take a trip w/a commercial outfitter, or use a company like canyon REO, or Moenkopi, these companies work with privates, and will do anything from your menu & food pack, to renting you a package, complete with gear, rigged boats, packed food, FD's, etc. But they aren't going to give you boatmen/women...you gotta take a commercial trip, or get invited on someone's private.
From your inquery, I'd save Grand Canyon for when you can make the time, and take a commercially guided trip in Dinosaur National Monument, or even better, the Middle Fork of the Salmon in Idaho. There are so many great, shorter trips....If you message me, I can give you some more ideas, once I know what, when, etc., you are looking for.
JJ