Diagnosis
You're right to be careful. The lenses have a multi-layer anti-reflective coating that can be damaged by rough cloths or cleaners. Dust and fingerprints are the usual suspects for blurry footage.
How to Fix It
The safest method is using a camera lens cleaning kit. If you don't have one, here's the shop-teacher approved method. 1) First, use a rocket blower or a can of compressed air to gently blow loose dust off the lenses. Don't use your breath—you'll spit. 2) Get a microfiber cloth meant for glasses or screens. Never use paper towels or your shirt. 3) Lightly breathe on the lens to create a faint mist. 4) Immediately use the clean, dry microfiber cloth in a gentle circular motion from the center outward. No need to press hard.
If That Didn't Work
If there's a stubborn smudge, dampen a corner of the microfiber cloth with a tiny drop of distilled water—not tap water and absolutely not window cleaner or alcohol. Wipe gently, then immediately dry the lens with the dry part of the cloth.
Heads Up
Never spray any liquid directly onto the camera lenses. Liquid can seep into the tiny seam around the lens and damage the internal electronics. Always apply the cleaner to the cloth first, not the device.