Many roads could be closed in winter, or if not, you're at least required to have tire chains in your car. Some people may not be used to driving on narrow, winding roads with the combination of snow -- especially for hours and hours. We made a bad choice for using California Highway 41 because it's the most mountainous I believe. California Highway 140 from Merced via the town of Mariposa is supposed to be the least mountainous and therefore, least likely to have the tire chain requirement. But all the major roads (41, 140, and 120) into the park are winding and can be stressful for the driver. So be prepared for that! Tire chains can be purchased or rented in the towns near Yosemite. Also, if you are a slow driver and are annoyed with impatient drivers following closely behind you, there are areas on the shoulder of the roads where you can stop and let them pass.
Plan your driving time well. There are no lights on the road at night and the roads get confusing. Get up there before it gets dark, or else you are driving in absolute pitch black... can't see 5 feet in front of you.. also this makes it hard to set up a tent/fire with only a flashlight.
Once you get into Yosemite National park, you will be treated to long, and winding roads, frequently with blind corners. Be very careful especially at night and when it rains. The roads can sometimes be quite narrow. I remember one nasty turn when we were driving up to Glacier Point, where the road curved sharply and if you were driving any faster than 20mph... well lets just say that there's no guard rail and its a long, long, long way down. So drive slow and carefully... there's no need to rush, Yosemite ain't going anywhere.
I will echo a bit what Al has said: Yosemite in one day will be no fun. Leaving from SF and driving by car or bus is a 6 hour round trip. June is a summer vacation month so the park will be a lot more crowded as will the roads going to the park. And you mention that you will have children with you also, it will be a long and tiring day for them and having raised 4 children they probably will not be pleasant at the end of the day.
Saying all that you will probably still go to see the sights and say you have been there. It is a magnificent park. My wife and I stayed outside the park for 3 nights back in October, 2010 and enjoyed it very much. To leave the hassle of driving that far just for one day I would do the tour bus for your family. I don't know anything about private tours.
Fill up your tanks around 50 miles out. We forgot and paid close to 75 cents more than normal. Gas in the park is expensive and not the easiest to find.
Yosemite roads can be very congested at times. There are more cars than the road was built to handle. If there are too many cars in one area, the rangers will close it off.
On my 1st ever visit to Yosemite in May of 2003..big ol blizzard at the 6000 foot level...made the RV trip alittle more interesting just go slow ...let faster idiots pass and enjoy the ride!!