Lived in Norcal for 12 years (Bay Area) with lots of camping trips to Yosemite, and fishing trips there and to other Sierra streams and rivers. In my experience, the easily accessible rivers in the park do not offer the best fishing.
Within the park, I think the lower portion of the Merced river (the pocket water and down through the gorge at the west end), the rivers in the Tuolomne meadows area, and the rivers off of the glacier point road, including the upper portion of the Merced river, offer the best fishing.
Outside the park, my personal favorite is the Stanislaus river, but the Tuolomne out of Hetch Hetchy is good as well. The Eastern slope is less well known to me.
I would recommend the Mother Lode fly shop in Sonora, as good with advice should you be traveling up from the Bay area...you should drive right past it as you are heading eastwards out of Sonora.
One of the most spectacular parks in the country....enjoy...there is also a book on fishing yosemite...I think from Amato publications...and a 5 page booklet that's available at all of the visitor's centers.