I lived at Tahoe for a couple of years, and have been numerous times on vacation.
If you want convenience, the state park in Tahoe City is right on the water, as well as adjacent to a shopping center that has Safeway, Baskin Robbins, a bakery, restaurants. This one has showers.
D.L. Bliss state Park is close to the water and is surrounded by state and/or national forests. It is just north of Emerald Bay. I think this one has showers.
Camp Richardson is on CA 89 just north of the South Y. It is run by a concessionaire, leased out by the forest service. It is in the woods, but there are so many spots it is a circus; there can be so much campfire smoke that it's hard to breathe. There are showers. There are a couple of very nice beaches within walking distance, several millionaire's mansions from the late 1800s, paved bike trails through the woods, and a nature center with a glass wall along a salmon stream (you go underground to view it).
Just north of Camp Richardson, on the road to Fallen Leaf Lake, is a (national forest-I think) campground.
There is a big campground in the middle of South Lake Tahoe. Very busy. You have to cross a busy main road to get to a beach that's not so great.
There is a small primitive state park campground at the entrance to Cascade Lake. No showers, very cheap, but maybe a 2 day max.
There are several inexpensive campgrounds along the Truckee River between Tahoe City and the town of Truckee. No showers, but quiet.
Just south of Tahoma is a campground in the forest: Sugar Pine Point State Park. Pretty quiet. There may be showers, but it is some distance to the water.
On Hwy 89 about 4 or 5 mi. south of Tahoe City, near the dock and restaurant called Sunnyside, there is a campground that borders the national forest. I think there are showers. I had a run-in with a little black bear there one day.
Meeks Bay has a private campground right on the water that's very crowded, but has hookups for even the largest Class A vehicles.
Kings Beach has a big campground right at the beach. Kings Beach is the armpit of Tahoe.
There's a big private campground just north of Zephyr Cove on US 50 on the NV side. Also on the NV side at Sand Harbor there is some camping. The 3 beaches are gorgeous, but quite crowded.
(Edit: correction-I think Sand Harbor is day use only perhaps.)
There are lots of unpaved roads all around Tahoe, going all back into the Sierra. The most famous, of course, heads due west from Tahoma: The Rubicon Trail
Let me know if you have questions.