There are two day hikes which come to mind aside from the one to Lone Pine Lake.
The first is Meysan Lakes trail, which is in the Whitney Portal campground- it's well-marked, but if you can't find it, ask Doug or whoever else is at the Portal store. This is a nice workout, though the trail peters out as you near the lake. It's VERY exposed to sunlight and brutally hot with almost no shade, start this one early. I would not begin this later than 7:00 am at the latest.
An even better workout is to start below the portal at the Lone Pine Campground. You'll see it on your left as you're driving up the portal road, it's right after you see the sign "Entering active bear area." Park in the far end of the campground and you'll see a marked trail beginning through sand. This climbs up and takes you to the Portal, where you an go on to Lone Pine Lake-- this is a great, little-used trail. Here is the URL for a good description:
http://www.californiahike.com/hikes/lonepinelake/home.htm