Good to have you as a member, look forward to your input. Yosemite is one of my favorites. For a first visit I would stay in the valley, there is so much to see and do. I think the fishing is just fine there, and early morning go outs can yeild fish, time to your self and kids, as well as the valleys bountys (deer, cyotes etc) You can get your licenses (over 16 years) right there at the store. Inside the valley is C+R for Rainbow Trout, and a 5 bag limit on Browns, so you could bring one back to have with your eggs if you choose, but I would do C+R on all, because of the pressure on such a small space, good time to teach the kids about the bennifits of C+R.
Artifical flies / Barbless hooks are the rule, so remember to pinch! Don't forget to set aside some time for a meal, or just a cup of coffee and danish at the Ahwahnee Hotel, a visit to the Le Conte Memorial, and a hike (sunrise is best) up to Yosemite Falls!
I envy you my friend, it's been a couple of years since I've been back, time to put it on the trip list. There are other Flyfishing destinations in the area, if you get time to yourself......But a man could spend a month in the Valley, and loose track of everything else!