By 6, my DS went on every ride at DL, however we had gone to DL many times by then, and he was very used to rides. The roller coasters are relatively mild, but many rides are dark and a little frigtening for some. I've gone with 7 year olds that had difficulty with many of the rides at first. It depends on the kids, and if they are used to amusement park rides. If they aren't, I would start them on a couple of the dark Fantasyland rides like Peter Pan to get them comfortable with dark rides. Those rides have low flow rates with slow moving lines so they are good to ride early. If they are comfortable with riding dark rides, they can graduate to Pirates of the Carribean and Haunted Mansion. The lines for these rides move faster, so they can wait for a little later in the day. If the kids have never been on a roller coaster before, take them on the small roller coaster in Toontown as early as possible to get them used to riding coasters. Again, this line moves very slowly so you need to go before the park gets crowded. They should like this quick little roller coaster, and then can graduate to Big Thunder Mountain Railroad and Matterhorn. You can get a FastPass for Big Thunder, but Matterhorn doesn't offer FastPass and so it should be ridden fairly early if possible. (If you don't know about Fast Pass, read up on it separately). Space Mountain is the most intense roller coaster at DL and might be a little intense for the little ones. It is reoopening in July after a long refurbishment, so it may be even more intense than before. You'll want a FastPass for it, and to ride Splash Mountain. Splash is a slow canoe ride, but can be a bit frightening for some kids because it has one massive drop. However, the kids can watch others going over the drop to see if they are willing to give it a try. Indiana Jones , though not exactly a "roller coaster" is another ride that can be a little intense, but it is very fun. Again, FastPass is recommended. This ride probably shouldn't be done until the kids are comfortable with thrill rides.
There are many other rides that are loads of fun even if the kids frighten easily.
Now I've only talked about DL, not California Adventure. Are you planning on going to both parks? DCA has a couple of rides that may be a little much for the 6 year old. Will you just be there for one day, or are you going multiple days? That could affect your strategy.
Lines will be very long in August, unless you are coming at the end of the month after most school starts. Use FastPass and start as early as possible. There is an information board at the end of Main Street on the left hand side that will list current queue times for the most popular rides; it's worth checking throughout the day.
I've let my DS go on rides on his own for a few years now (he's now 14), but he is very used to DL. We'd also use radios to keep in touch. If you do separate, make very sure you know where and when you will meet up. (The information board with queue times may help you figure timing) I don't think safety is an issue, but the parks will be crowded, so it there is a risk of losing track of one another, and that can be very stressful. Another thing you can do is "the baby swap". You all wait in line together, and then when it comes time to ride, one of you stay off to the side with the kid(s) that don't want to ride. After the first one finishes their ride, you switch off. This can also be helpful if one of the kids chickens out at the last minute.
If you are staying multiple days, I defnitely recommend taking a break in the middle of the day. Otherwise, just find some down time at the parks midday having a restful meal, watching a show, etc. And be prepared to be tired...
Keep reading the boards to learn more. And have lots and lots of fun.
PS -- I agree it's too late for a stroller for most kids by this age.
