I spent 8 days fishing just the park a few years ago. Access is not a problem as the vast majority of rivers are road accessible. One thing you should know is that it is not uncommon for some of the rivers (firehole, Madison, lower gibbon) to be closed in the summer (august) due to water temps. If you go in july shouldn't be an issue.
The rivers I fished in the park:
Yellowstone: Sucked in august. I fished it outside the park in paradise valley a couple years later and it was good
Gibbon: Pretty technical meadow fishing. I liked it a lot. My brother got a 22inch brown.
Lamar/soda butte: Good cutthroat fishing
Gardner: Good pocket water fishing for medium sized browns and rainbows
Slough: Good cutthroat fishing. Unfortunately the upper meadows were closed when I was there due to grizzly activity. Otherwise I would have loved to hike into the upper meadow
Snake: In the park was just ok. Better sections outside the park.
There are a ton of other small streams. I fished a couple of them but don't remember what they were called.