Judging by the way you posed this question, this is likely your and most of your groups first trip. You will want to hit most of the highlights, depending on your abilities and how the trip lines up, here is my list. It's a big place, so you may not hit everything, and that's ok too:
North Canyon (nice pool here. short)
Silver Grotto (some climbing experience needed)
South Canyon (ruins, narrows, views)
Redwall Cavern (not a hike, really a stop for some Frisbee. Bring a Frisbee)
Nautiloid (worth it if you camp here really)
Buck Farm (Nice hike to some narrows)
Eminence Break (good views)
Saddle Canyon (longer hike to a cool falls and narrows)
Nankoweap (granaries, up the creek, etc.)
Carbon/Lava Loop (Great morning hike with a boat shuttle)
Cardenas (Go up to the Hilltop Ruin)
Unkar Delta (Self-guided tour of ruin sites)
Tabernacle (From Rattlesnake Camp, a good moonlight hike too)
Vishnu (Hike up over the fin into some nice Schist, watch the pull in here, easy to miss!)
Clear Creek (Very Schisty here)
Monument Creek (Granite)
North Bass Area from Bass Camp into Shinumo Creek
Tapeats/Deer Creek Loop with a boat shuttle (Map on front of Tom's guide)
Deer Creek to the Patio/Throne Room
Olo (Good climbers only to get in here)
Havasu Creek (Plan 1/2 day here)
Tuckup (Great narrows, close to the River. Lots of options here)
National (One of the nicest side canyons off the river)
Fern Glen (Nice camp with a cool hike up to some falls)
Below Lower Lava Camp (Stop here and look for dead brain cells. Pretty easy to spot)
Lots of folks use these as mileage days, but there's still places to stop, and still some bigger rapids below Parashant:
202 Mile (Some petros up this way)
Walk down and have a look at Pearce Ferry Rapid after the take out.
There are lots more hikes, and some folks may not want to do everything. Hope this list helps. I am a big fan of spending a full day between Tuckup, National and Fern Glen. Such nice Muav canyons. Nice easy day before Lava too.
You would likely have time to see everything I listed above and more.
Have a nice trip!