Where exactly are you going? There are two kinds of redwoods in California and they grow in different places. Both are tall, ancient and spectacular. No matter how old you are, you'll feel like a tiny young kid when you're standing next to a giant 2000 year old tree. The key to enjoying them and learning about them is to get out and walk. Find a trail map and hike away from parking lots, roads and campgrounds. You'll find yourself in places that literally are like nowhere else on earth. Some state parks have trails for the disabled that are flat. Some have trails that are paved and made for wheelchairs. Of course, there are more rugged trails, as well. From inside a vehicle, it's impossible to feel the power of a forest of 100-300 foot tall trees.
The coastal redwoods (Sequoia sempervirens) grow from around Santa Cruz to a little bit north of the Oregon border. Because they grow along the coast and thrive on foggy weather, you can probably expect temps in the 50s-70s. Typically, the fog blows inland in the late afternoon, stays overnight and burns off mid-morning. Wear layers of clothes. When the fog burns off, it can warm up at least 10 degrees fairly quickly. When the fog moves inland, temps can drop quickly, as well. A tent, as long as it's out of the wind and you have a rainfly, should be fine. Why do you need a rainfly, even when it's not raining? Redwood needles are arranged on their branches like combs and they "comb" the wet fog to get water. That water then drips down onto the forest floor. The trees and the fog can drip all night long. That's why you'll need the rainfly on the tent.
The redwoods in the Sierra Nevada Mountains are a related tree, the Giant Sequoia (Sequoiadendron giganteum). Some of the most famous of these grow in Yosemite's Mariposa Grove but they grow along most of that mountain range. Because they grow at higher altitude and rely on snowpack and storms for water, the weather there is akin to the weather in the North East U.S., except that the summers are *bone* dry and as a result, are prone to fires. In May and early June, you may get cold temps (upper 30s), especially at night, along with occasional storms. A tent is a maybe in that climate. The issue that time of year in Yosemite won't be the weather, it will be the crowds. Unless you have a reservation, forget it. There may be campsites available in the surrounding areas, but again, without a reservation, you may end up with no place to stay.
You'll probably see various locations described as having "old-growth" trees. That's a forest that's never been logged. A second-growth forest has been logged but generally, that happened about 100 years ago. If you can, visit an old growth forest. You'll generally see larger, older, more majestic trees in mature forests. In May or June, you might be too late to see the native rhododendrons and azaleas in bloom in the coastal redwoods or the white dogwoods in the Sierra Nevada redwoods, but there are plenty of other beautiful plants that are a part of a mature redwood forest.
For information about California's State Parks, visit:
http://www.parks.ca.gov
For information about California's National Parks, visit:
http://www.nps.gov/state/ca/index.htm