Deb covered a bunch of what I was about to mention. I'll leave most of it for emphasis.
http://www.summitpost.org/eastern-sierra/154084Do test out how you feel at 12000, 13000, 14000, if conditions permit, but don't stress about the acclimatization and where you sleep if you are patient enough to save Whitney and any other 14ers for later in your trip. You'll know if you'll be fine by then and likely be well adapted. There are plenty of camping opportunities in the 8,000-10,000 foot range to help you adjust to altitude. Sleeping high may not matter if you are patient about when to do 14ers and go at a turtle pace on those hikes. Furthermore, don't be fooled into thinking that 14ers are more fun and scenic than 12ers and 13ers. White is less than exciting. Langley is less than thrilling by easiest trail route. Whitney's easiest trail route is more interesting than White and Langley, but the Mountaineers Route and some of the other mentioned peaks in the Whitney area are much more exciting if within your abilities and gear under whatever the conditions end up being.
White Mountain is a good suggestion as an easy acclimatization test and experience. It is more barren than the Sierra, but does have the bristelcone pines to check out along the drive to the trailhead, has great views of the Sierra across Owens Valley, is a unique experience, is less crowded, is less likely to have much snow, and is a safe hike on a road. It is a far longer, rougher driving experience than Sierra trailheads, but if not snow covered, is a 2WD road (closed if much snow). Don't be tempted to race the dirt portion in a rental car with no spare and end up with a flat. Grandview Campground up there is pleasant and somewhat high, but a solid hour drive or more from your hike and be aware it has no water.
You will be there a week, are looking for dayhikes, are skilled scramblers, and if the snowpack is paltry enough there could be several other worthwhile Sierra peaks or hikes not mentioned yet that you could fit in. May is less crowded than summer, but I personally tend to avoid the crowd and permit hassles of the Whitney zone. Don't get me wrong, do Whitney and some of those other peaks near it; the area and those peaks are great, just not necessarily better than less famous areas. As mentioned, check out the Inyo National Forest Whitney permit lottery online right away. Langley (mentioned) while not as exciting, is a good warm up for Whitney and tends to have less snow than places farther north. Also likely to your advantage, it has little steep, north facing terrain via New Army Pass. That hike has lots of flat terrain. The Horseshoe Meadow campgrounds are good for acclimatization.
Mountaineers Route description can be found in the Mountaineers Route folder after clicking into the main Whitney Portal Store Message Board Section (the one at the top). Also in that message board are general info. and condition reports:
http://www.whitneyportalstore.com/whitney-messageBoard.htmlWebcam Peaks and MR landmarks labeled:
http://www.whitneyportalstore.com/wc_notes.htmAs mentioned there are great areas north of Whitney, but chances of deep snowpack and storms generally increases as you move north. May is usually too early for an easy trip.
Onion Valley campground and trails are high, good for acclimatization, and beautiful. Kearsarge Pass and Gould are easy. University and Dragon are more challenging, but probably within your ability if conditions allow. Beyond those, ambitious and fit hikers can get to stunning areas a bit deeper into Kings Canyon NP in a dayhike, but not if slowed by snow. Ask or research if interested.
North Fork of Big Pine Creek brings you views of the Palisades!
South Lake trailhead out of Bishop has great peak opportunities like Agassiz, Goode, Chocolate, and Cloudripper. Bishop Pass and Dusy Basin is a scenic hike even without tagging peaks. Nearby, Sabrina trailhead and North Lake trailhead have excellent hikes. If already open, North Lake campground is high and nice. Piute Pass to Muriel Lake area is scenic. There are peaks to tag if up to it. Evolution Canyon over Lamarck Col. is stunning. Mt. Lamarck could be tagged.
Rock Creek Road brings you to stunning high altitude trailheads, campgrounds, hikes into Little Lakes Valley, Ruby Lake, Mt. Starr, and Mt. Morgan, not to mention harder classics.
Farther north, there is the stunning Banner & Ritter area, but you are probably too early in the season for that (lots of snow most years, but maybe not this year). It would be a similar situation for Dana, North, Conness, Excelsior, and Tuolumne Meadows area near Tioga Pass, as well as Matterhorn Peak near Bridgeport (there are nice hot springs near a quiet camping area if you venture that far north). These are mostly 12ers or low 13ers, but are great experiences that hold their ground against higher peaks to the south.
All these peaks have Summitpost mountain pages and most have specific area overview pages available as well. On those pages you can see all information you should need. Inyo National Forest website should give you what you need for camping information. The books are great, but not necessary. Also try highsierratopix.com if you want a forum discussion or search, but be aware advice tends to be on the safe side there and more forthcoming if you state your experience and abilities as well as show realization that May tends to be snowy on the ground and it is still too early to know the snowpack and weather.
Maps can be printed from Summitpost, CalTopo, snwburd.com/bob, and other places. If wanting to spend money on maps of general areas, National Geographic makes waterproof maps or Tom Harrison maps tend to be well respected. National Geographic map would be #205: Sequoia/Kings Canyon NP, and #809 Mammoth Lakes Mono Divide if heading to Bishop. Way north, #309 and #308 for eastern Yosemite area.
Edit: this new thread is worth a look if you haven't seen it
http://www.summitpost.org/phpBB3/viewtopic.php?t=69018