For your Sony A6000, a versatile lens that would be suitable for normal shots and YouTuber-style videos within your budget of $300 is the Sony E 50mm f/1.8 OSS lens. This lens offers a "normal" focal length on an APS-C sensor camera like the A6000, which means it closely resembles the field of view of the human eye. Here are some detailed reasons why this lens is a good choice:

1. Focal Length: The 50mm focal length is considered ideal for general photography, including portraits, product shots, and everyday situations. It provides a natural perspective, neither too wide nor too narrow, making it versatile for various shooting scenarios.

2. Aperture: The wide maximum aperture of f/1.8 allows for excellent low-light performance and shallow depth of field. This means you can achieve a blurred background effect, making your subject stand out while maintaining sharpness.

3. Optical SteadyShot (OSS): The lens features built-in image stabilization, known as Optical SteadyShot, which helps reduce camera shake. This is particularly beneficial for handheld video recording, resulting in smoother footage.

4. Image Quality: The Sony E 50mm f/1.8 OSS lens produces sharp images with good contrast and color rendition. It also minimizes aberrations and distortions, resulting in high-quality visuals.

5. Compact and Lightweight: This lens is relatively small and lightweight, making it easy to carry around and suitable for vlogging or travel videos. It won't add much extra weight to your camera setup.

6. Examples and References: You can find sample images and videos shot with the Sony E 50mm f/1.8 OSS lens on photography websites, YouTube reviews, and forums. These resources will give you a better idea of the lens's capabilities and help you assess if it meets your requirements.

Overall, the Sony E 50mm f/1.8 OSS lens is an excellent choice for normal shots and YouTuber-style videos. Its versatile focal length, wide aperture, image stabilization, and compact size make it a popular option for many photographers and videographers.