Welcome touron! California is big. Really, really big. And hard to access. 10 days is not enough, unless you want to just drive the whole damn time. If you want to camping in June at a campground, you needed to make reservations in Feb.
Logistics: coming to CA via car in mid-June. Limited options to enter/exit the state. 5 to the North to OR, 80 and 50 via Tahoe, 15 via Vegas, or 40, 10 and 8 through the desert. Odds are very high 120 via Yosemite will be closed, it didn't open until June 29 of 2017, and this year is bigger than that. The Eastern Sierra via 395 is beautiful, but it is cutoff and Yosemite is a dead end instead of a shortcut of sorts.
Here's my credentials: I'm born and bred CA. My folks were big on seeing all of the state when I was growing up, so we did. Everything listed is something I've done.
FYI: We love pets and I'm sure your dog is awesome, but our tourist attractions by and large are not dog friendly, including national parks. Your dog is a big issue. Assuming you ditch the dog, here's a big list of things that might be interesting
Socal - Has it all, except nature because there's 20 million people there and its all paved
Disneyland - a classic for kids of all ages. Your kids might not think they like Disneyland until they walk in. The rat is the master of the craft and will eat your whole paycheck. And you'll be happy the whole time as you pay for it all.
Magic Mountain - arguably top 3 destination for roller coasters on the planet if that's your thing
Universay Studios - Harry Potter land is pretty damn good if that's your thing.
Knott's Berry Farm - Solid 1 day amusement park. Might get shanked, they had some security issues last summer
Legoland - probably too young for tweens unless they're hardcore lego fans, but really cool if lego is your jam
Any Orange County Beach - it's exactly what you'd expect, large open beaches that are nicer than LA County beaches. Lots of room. No parking to be found after 10am. Get a surf lesson for lots of fun.
Venice Beach - an absolute mismash junkshow scene of tourist trap, artists, street performers, junkies, and hollywood posers hoping to get noticed. Fun for a while to take in the scene
Griffity Observatory - SPACE
Getty Center - World famous art museum
San Diego Zoo - Absolutely world class facility
Tacos - everywhere
Fish Tacos - almost everywhere
LA Dodgers - Sit in the bleachers and get in a fight
La Brea tar pits - if you have a Dinosaur loving kid, this is mecca
Hollywood - you can see just how thin the glitz and glamour veneer really is, it's actually a bit interesting if that's your thing
Thousands of great restaurants, music venues, and so much more.
Deserts
Joshua Tree/Mojave/Death Valley - check up on your AC before you depart CO and just see them while driving. Don't get out. Not in June. Hot by 8 am, deadly by 10.
Needles/Lake Havasu - If you someone with a boat on the river, it's actually a good time.
Central Coast
Highway 1 from Morro Bay to Monterey - a beautiful drive, breathtaking views. But the road currently doesn't go through due to landslides and repair work, which makes the central coast another bit of a dead end location. Not thilling as an 11 year old.
Hearst Castle is beautiful and a bit historic, but a bit out of the way for a fancy house. If you can manage to get in on a swim in the Neptune pool, do that for sure and be Gatsby.
Monterey Bay Aquarium is world class, kids that love the ocean will spend a day here. Get in very early before it gets crowded, and go check out tidepools in carmel in the afternoon. Cannery row is a tourist shithole, gtfo. Dennis the Menace park is an all-time playground for the kids, but they might be too old. It's free and dog friendly
Santa Cruz Boardwalk is good cheesy beach carnival fun. Rides, beach, pier.
Laguna Seca - awesome racetrack even just for simple events
Pismo Beach - Drive around and camp on the beach, or rent a quad and go be a redneck on the dunes! BRRRAAAPPP BRRRAAPPPP!!!
Vandenburg AFB - If the timing works, maybe see SpaceX/Blue Origin shoot one off
Central Valley
There actually are a few good things here, just nothing worth your short time. Once we have a better idea of that you want out of a trip, stops can be added as you pass through
Central Sierra
Yosemite - Yes, it will be crowded. Yes, the waterfalls will be all time and I'm planning on going in June/July because it is going to be all time. Yes, just about every trail in the valley is paved so you can take the dog. No, Tioga road to the Eastern Sierra will not be open by July 4th this year, probably later. It will still be absolutely amazing, there is no place like this on Earth. I've been at tunnel view with 10 buses of Asian and German tourists jamming the parking lot and it's still a place of wonder and amazement. Just go. I think Mariposa Grove will open late this year for road repair, so you might not get the big trees here.
Sequia and Kings Canyon - Cut rate Yosemite, still great, but it's not Yosemite. More big trees. Will still be crowded with all the people that thought Yosemite would be too crowded so they went here. The secret is out. Just go to Yosemite instead.
Eastern Sierra
Most hiking is likely snowbound this year and sadly not worth the detour as it's isolated
Mammoth will be open and is IKONIC if you need to try the CA snowpack.
Ancient Bristlecone Pines are otherworldly, but likely closed due to snow this June
Manzanar is a historic reminder that your grandparents are racist
San Francisco/Bay Area
Ride a cable car
Exploratorium is a great kids science museum
Watch a bum shit on the sidewalk
Walk across the Golden Gate Bridge
Muir Woods if you can get in. It will be crowded, but it will scratch the giant redwood itch without burning a whole day or two to get there
Acadamy of Sciences is another great science museum
Catch a musical on Market St
Buy hard drugs on Market St
Catch a Giants game in a really nice luxary stadium for some big bucks
Catch an A's game and piss in a 30' long trough inside a momument to concrete for under 10 bucks. See why the Raiders left and the A's are moving to Vegas.
Napa Valley/Wine Country - not for kids
Great America/Six Flags Vallejo - Norcal has roller coasters too! Just not as good as Socal
Sears Point - another good racetrack even for basic weekend events
Sacramento and Sac Valley
Old town Sac and the Train Museum can scratch the history lesson itch for the wild west
Pan for gold up in the foothills
Downieville for mountainbiking
More later. It's a big state.
Wait, how can we trust this guy^^^ He's clearly not DJSapp
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