Any ideas about how we can plan a visit to the United States?

I love Yellowstone, but you still have to get there.  If you are flying to the East Coast you'll have to then fly to probably Denver and then drive to Yellowstone. You are looking at least 2 days each way which will cut your trip short. A couple of friends and I drove from Minnesota to Yellowstone (2 long days each way) and then we camped, I think we spent about 10 days in total on the trip but it was a hard 10 days and honestly, I'm not sure I'd want to do it with young kids. 

I hate to say it but you don't really seem to grasp how large the US is, especially the western states.  Before I moved over, my husband and I did a road trip from Minnesota to Ontario and it nearly drove him insane. Took us 2 very long days and even the beautiful northern coast of the Lake Michigan got really old after an hour (though we did find a chap runing a little petrol station in the middle of nowhere that heard my husband and started speaking Welsh to him!)

If you did want to do Yellowstone, it would probably recommend it be ALL you did. I still think something like New England or the Pacific NW would be your best bet if you want to go to more than one place, unless you want to spend the entire trip traveling which will get really boring for the little ones. 

I hope you let us know what you decide to do. 

gotta correct--as one who lived there.  If you want to look at Yellowstone, the closest major airport will be Salt Lake City, not Denver.  It's then about a six hour drive from there.  I love the area, you could easily spend a huge portion of the whole time in the region--Yellowstone, Grand Teton National Park, Jackson Hole, some great waterfalls and canoeing in Eastern Idaho (Island Park has some amazing hidden gems).  It's also family friendly with good nature, lots of choices from nice hotels to camping or cheap little lodges.  You probably won't see quite as much wildlife in August as you do in the spring, but it's still amazing.  I really miss being there throughout the summers.

Being a West-Coaster, I'd say the only South that's safe to visit in August is Southern California  You can't go wrong.
I assisted my English boss in planning a trip up with his family (teenage children) along the California Coast, from San Diego to San Francisco, and then including Yosemite National Park.  They LOVED it.
I love NE, but in the Autumn.
I love the SW, but in the Spring (my family and I just did a trip around AZ, NM, CO and UT, and it was stunning - but again, before it gets too hot!).
California has everything to offer - from amusement parks, to the most beautiful beaches in the world, to fabulous mountain views, National Parks, and culture.  And 99.9% guaranteed good weather.
DO NOT SPEND TIME SEEING 'L.A.' - it's a big city.  No big whoop.

And yes, if you have more time, definitely head up to the Pacific Northwest.  More gorgeous country to see.

When I was very young my family would take road trips from Southern California up to San Francisco, over to Yosemite and Sequoia... all doable with young children, and plenty of clean, inexpensive places to stay along the way.

As above, if you're going to do something like Yellowstone, then keep it at that.

For Yellowstone - Would it be possible to fly into a regional airport on a connection from Denver or Salt Lake City?  Or is that not done, or too expensive?  Looks like there are some smaller airports closer in.

DO NOT SPEND TIME SEEING 'L.A.' - it's a big city.  No big whoop. 

Oh, but I loved LA!    (An old friend lives in WeHo.)

Mann's Chinese Theatre, the Hollywood Walk of Fame, Sunset Blvd, Santa Monica (the pier & the beach) & Venice Beaches, El Capitan, Mulholland Drive, whatever that area is famous for rock-n-roll & all the leather/interesting shops (is that Sunset Blvd?  not been there since 1992-93), was there when the Academy Awards were on, smell of jasmine in the air, and then I went to Disneyland!    (Didn't love the traffic, though.)

gotta correct--as one who lived there.  If you want to look at Yellowstone, the closest major airport will be Salt Lake City, not Denver.  It's then about a six hour drive from there. 

I wasn't sure which was closer, but I knew it was still a drive from either one lol

For Yellowstone - Would it be possible to fly into a regional airport on a connection from Denver or Salt Lake City?  Or is that not done, or too expensive?  Looks like there are some smaller airports closer in.

Sure you can, I was just thinking of saving expenses.  The regional flights are hit and miss--sometimes really cheap, sometimes really expensive.  There's not rhyme or reason to it from what I could see.  Idaho Falls (where I used to live) has a reasonable regional airport and is a pleasant drive from Yellowstone--only two hours with fun stuff along the way.  Jackson, WY also has a regional airport which puts you right at the foot of the national parks.  I don't know on prices as I always drove from home, but I imagine they are expensive as it's used by lots of celebs and such and you have to fly into the mountains.

Oh, but I loved LA!    (An old friend lives in WeHo.)

Well alrighty then - I take it back!  Depends on what it is one wants to see then.  I just know of too many people who were so excited to see 'Hollywood' and were sorely disappointed.  But you're right, it does have its offerings.
However, DISNEYLAND, which is indeed the Happiest Place on Earth, is not in LA!  Just had to make that distinction

It can be really hard visiting LA because of the TRAFFIC. UGH! Driving around, with it all hot out, not knowing which of the million freeways I should be on...

But, we did have a good experience last time. I took my boyfriend to Universal Studios on election day (I had already voted via post from England), and NO ONE was there!! We ran around on every ride, several times.

Next presidential election, we're totally doing Disnelyand...

I love NE, but in the Autumn.

Summer in New England is really lovely! Clamcakes, frozen lemonade and picnics on the beach!

However, DISNEYLAND, which is indeed the Happiest Place on Earth, is not in LA!  Just had to make that distinction  

Oh yeah, I know!  But it's not far away & in my mind, it was part of my overall LA experience! 

Also I loved La Placita Olvera, and the Capital Records Tower. 

Summer in New England is really lovely! Clamcakes, frozen lemonade and picnics on the beach!

Yes, I should have phrased that differently... I loved NE when I was there in Autumn... have not experienced it at any other time!

Oh yeah, I know!  But it's not far away & in my mind, it was part of my overall LA experience! 

Also I loved La Placita Olvera, and the Capital Records Tower. 

You're reminding me of all the places I need to revisit, thank you!

Yes, I should have phrased that differently... I loved NE when I was there in Autumn... have not experienced it at any other time!

I'm a bit biased. 

You mentioned flying to the East Coast, where will you fly into?  Obviously flying to Miami or to Boston or somewhere in between will dictate what sort of places to visit if you are only planning to drive after that.

If you are going out West you'll have to fly and you said you were budget conscious so that could cost a fair bit extra, not to mention the hassles of flying after going trans-Atlantic.

I think your best best is to fly to Boston or New York (the latter could be a fair bit cheaper possibly).  That gives you New York and New England to explore.  Alternatively, you can fly to Baltimore or DC and do these cities and places like Skyline Drive along the Shenandoah valley or head to Ocean City or Virginia Beach.