If you don't want to backpack then the only way you'll be able to climb Mt. Whitney is by accessing it from the eastern side of the park via Whitney Portal. Just keep in mind that it is an extreme day hike; 22 miles round-trip with over 6,000 feet of elevation gain, and that's starting from an elevation of over 8,000 feet. It also will take about 6 hours to make the drive from Whitney Portal around the south end of the Sierra Nevadas, and back up into the main part of the park. There are no roads into the park from the east. Here is some more information on the Whitney day hike if you're interested www.dayhiker.com/directory/MtWhitney.htm
In the main part of the park, some good hikes to look into would be the Alta Peak trail, the Lakes trail, and the Twin Lakes trail. These are full day hikes but there are many shorter hikes that I typically suggest as well, such as the Moro Rock trail, the Crescent Meadow loop, and the Congress trail. There are over 800 miles of trails in the park and many more hikes in addition to these, but hopefully this helps get you started.