The world's tallest mountain, measured from the ocean floor, is not Mt. Everest! Instead, it is Mauna Kea, a dormant volcano in Hawaii: it measures more than 10,000 m (about 33,000 ft, or 6 miles) from the ocean floor.
To native Hawaiians, the summit is a spiritual place where only royalty and tribal chiefs were allowed, but now, thousands of visitors go there each year in addition to those visitors that go for spiritual purposes. Driving to the top is a possibility, but Maunakea is one of the only places in the world where you can drive from sea level to 14,000 feet in about 2 hours, so altitude sickness is a high possibility. At 14,000 feet, there is 40% less oxygen than at sea level, so visitors should acclimatize to the altitude before proceeding further up the mountain by stopping at the visitor center.
The visitor's center is open from 9 a.m. until 10 p.m. 365 days a year. Informational multimedia presentations, souvenirs, and some food items are available here, as well as clean restrooms and drinking water.
One of the most amazing parts of visiting the mountain is that very night, regardless of weather, the Visitor Information Station at the Onizuka Center for International Astronomy holds a free public stargazing program. If you love Stargazing, this is somewhere you absolutely must go. Currently there are 13 independent multi-national astronomical research facilities located on the summit. While the summit is open to the public, the summit telescope facilities are closed to visitors. Check out the video on the 5th slide to see a beautiful look at this volcano!
If you want to splurge, book a room at the Shipman House Bed and Breakfast and relax in the sunny, privately owned Victorian mansion, complete with more than 20 tropical fruit tree varieties blooming in the garden: now that, is the Hawaiian lifestyle!
If you're looking for a more "earthy" experience, you don't need a permit to hik there. There is no fee. There are no camping facilities and only porta-potties are at the top. Lake Waiau is considered sacred and Native Hawaiians very much care about it.
If you want to acclimate, the place to do it is by camping at Mauna Kea State Recreation Area at about 6000 feet. For that you would need a camping permit. These sites have all the info: www.hawaiistateparks.org/camping/hawaii.cfm, http://www.hawaiistateparks.org/camping/ but chances are you would be staying in a cabin if you do this; I believe tent camping is illegal in this area. It costs $80 a night for non-residents.
To find more local trails around the area, visit this link: http://www.ifa.hawaii.edu/info/vis/visiting-mauna-kea/local-trails-around-the-vis.html