So, Big Pine is a small town with some Big Problems. I've shared that in an article on our companion blog, Glam on the Lam. You can (and if you plan to spend much time in Big Pine you should) read that here.
But there's some good, easily accessible, high-yield fishing if you know where to go.
There's a great little spot just south of Big Pine, which (in my experience) is a perfectly safe and really fun place to fish with kids because they can get what we call the fishing "Grand Slam," or 5 different species of fish in a very short period of time. We are sworn to secrecy about this spot, but our favorite Eastern Sierra guide who brought us here the first time, Otis, can take you there if you give him a call at 760-937-1865
In town, there's a some easy, fine trout fishing at the twin pools at Baker Creek Campground. In many situations I drop my kids off so that they can fish on their own time. I would not recommend doing that at Baker Creek as I can't be sure that this campground is a safe place for kids or teens alone. Luckily, it doesn't take long for kids to catch a ton of fish so its an easy, supervised stop on your way through Big Pine.
The turnoff to Baker Creek Campground is just south of Glacier View Campground, on the west side of the 395.
Take the Baker Creek turnoff and head west, following the signs to the campground. At T entrance toi the campground, turn left and then follow the curve to the right. You should see the two pools on your right, with a small, single lane road between them, leading back into the campground.
Fish on either side, but we generally have better luck at the pool further to the west.