When I visit cemeteries, I like to visit early in the morning or late (near sunset) because that's when the light is the best. If you do find yourself trying to take photos of headstones in harsh lighting, use your body or an umbrella to try to block the light so you don't end up with overexposed images or harsh shadows that make the photo unreadable.
Even if you can't read a tombstone, take as many photos as you can from different angles because you may be able to bring out details in the photos and/or get some help from your genealogy friends on the Internet once you've got the photos uploaded.
Possible things to take with you in your cemetery kit:
/// Gardening gloves for clearing away leaves/mud/etc.
/// Gardening shears for trimming grass and brambles
/// Spray bottle of water to make illegible headstones easier to read
/// Dry clean paintbrush to gently brush away dirt
/// Bug spray or protection against ticks and other cemetery inhabitants
/// Small flags, flowers, pennies, or other tokens you want to leave on the markers
/// Chalk for marking the road or trees so you know where you've been
/// Extra camera batteries or a spare charger for your phone if you're taking cell phone photos
/// Optional: aluminum foil if you want to do very gentle rubbings to make markers easier to read
/// Optional: flashlight if you want to try to create some artificial shadows to make markers easier to read
/// Do not take: shaving cream, harsh soap, dish soap, crayons for rubbings, or anything else that may harm the headstones.