Avatar automatically scales every image you upload to fit within their 'containers' so there's no specific set of dimensions we recommend. This applies to all images you upload into the theme: slideshow, logo, products, and collections.
For example, if you choose to display two products per row on the collection pages then the images will expand and display a lot larger than if you choose to display five products per row.
For the product images, we recommend images that are around 1800px on their longest edge. So, 1800px wide for landscape-oriented images or 1800px tall for portrait-oriented images. Avatar will create multiple thumbnail versions of the images and always use the smallest one possible that still fills the available space. Full-sized product images are only ever loaded in the 'lightbox' galleries that open when the main image is clicked from the product page.
For the desktop banner, images should be at lease 1200px wide in order to fill the width - images will scale down, but not up- but the height can be whatever you think looks best, we recommend between 400px and 600px.