I'm planning on backpacking, but I don't want to get in too deep.
Trails out of the valley are extremely well marked, so you can't go wrong.
Main trail is up the "giant staircase" to little yosemite valley to party with the bears. Can continue from there to Half Dome.
Less used is trail to top of Yosemite falls.
Suggestion:
On your 4-day schedule, you'd have time to drive out of the valley and up onto Tioga road, for an overnighter to May Lake.
Map
Before departure, ask the ranger station in the valley if you need a wilderness permit.
May Lake is spectacularly beautiful. I've been to all the camps out of Tuolomene Meadows many times, and rank May Lake as one of the most beautiful.
Me there
It has a full-on camp with tent cabins, showers, etc., but plenty of area away from all that for backpackers, too. All the other camps are accessible only by backpack or mule, but May Lake is just 1.5 miles off the road, all up.
From May Lake walk up Mt. Hoffman from where you can look down into Hetch Hetchy valley, twin of Yosemite.
First buy sack lunch at deli at store in the valley, drive up, hike in, camp. Absolutely do not leave food inside your parked car, as steel car is nothing to bears. Put food & scented things into free bear lockers at parking lot.
Best time for Yosemite is from last week in April to mid-May, when it's still really cold, before school lets out & crowds begin, before bugs hatch, while snowpack has started melting & waterfalls flow full.
I hope you have fun & post some photos of your trip here!
Liberty!