Tanner to Grandview would be a nice 6 day trip. This is more remote and you will see less people. Because you must filter directly from the Colorado river, make sure your water filter is field cleanable or else take a spare if you do this option.
Grandview to Hermit will involve more people and is also a nice trip. You will see more people and may have a tough time getting a permit for Indian Gardens so apply early.
If you have two off-road capable vehicles, you could probably do Bright Angle to South Bass, but the Gems are remote and water can be tough to come by in April if it was a dry winter. You also need to arrange a substantial shuttle possibly having to pay the Havasupai tax twice.
I would recommend sticking to existing trails for your first time in the canyon. While there is a lot to do off trail you would want more experience before attempting most of them.