I have looked in the player handbook and it does not seem very clear on character vision. In Bright light how far can characters see? For example if I have a player with the feat sharpshooter and have a long bow I can shoot 600'. Is there anywhere that talks about how far you can see in bright light? Dim light? Also for example if a character can read lips it doesn't specify distance away in which this is possible.
The 5e player's handbook does not have rules on visibility, but the DMG does. They can be found on page 243, under the "Visibility Outdoors" heading.
The basic summary of it is that, on a clear day, you can see 2 miles, or until your view is obscured by terrain features. If it's raining, it usually cuts it down to 1 mile. Fog cuts it down to 100 to 300 feet. And standing on a mountain or similarly elevated place will give you a visibility of 40 miles. For comparison, the horizon on earth is 2.9 miles away at ground level.
Dim light isn't mentioned as diminishing your visibility (beyond giving disadvantage on Perception checks), so dim versus bright light doesn't change anything.
As for the range at which you can read lips, there's no specific rules on this. Ask your DM if you can read the lips, or if you need to get closer.