Some must see/do attractions in Sequoia are the General Sherman Tree (largest living organism on the planet), the 2 mile Congress trail which begins at the base of General Sherman, the Crescent Meadow loop trail where you can see Tharp's Log (a cabin built into a fallen giant sequoia by the first non-Native American settler in the parks, Hale Tharp), the trail to the top of Moro Rock, Tunnel Log (which you can drive through), and Crystal Cave. These attractions are all located fairly close together in the Giant Forest area, and there are many other great trails there as well.
You are certainly welcome to bring a bike with you but there are not many places in Sequoia where you are able to ride, so that would be mostly for the Yosemite portion of your trip. There are bike rentals available in Yosemite Valley though, so you may want to just take advantage of that.
As for campgrounds, Lodgepole is the one that I generally recommend in Sequoia. It's right in the heart of the park and only a few minutes drive to the major attractions. They are also one of the only campgrounds in Sequoia that take advance reservations (go to www.recreation.gov to reserve). The only other campground that takes advance reservations is Dorst, which is about a 15 minute drive north of Lodgepole. All the rest are on a first-come-first-served basis.
There are several campgrounds in Yosemite that take advance reservations, the largest of which is Upper Pines. The other options are Lower Pines and North Pines, as well as Camp 4 which is a first-come-first-served campground. I would absolutely suggest making your Yosemite reservations as soon as possible. Yosemite is an extremely popular place, especially during the summer, and I'm sure they will sell out well in advance. You can book those campgrounds at www.recreation.gov as well.
Hope this helps!
Edited: 11:13 pm, March 08, 2013