In many cases, you can find recommended image dimensions throughout the theme settings for the various areas where images are required.  For cases where this information isn't provided, or if you'd like to access a single resource for this information, here are some basic rules of thumb for image sizing:

Homepage sections

Slideshow banners - 1400 x 500-800 px.  Height doesn't matter too much as long as there is room for all the text you want.  All slides should be the same size.  If you are using full screen slides in Galleria for the column slideshow or hero, we suggest 1600 x 1200px for these images.

Gallery featured images - 800 x 600-1000 px - Again, height doesn't matter much as long as all images in the same row are the same height.

Rich Text/Video background images - 1400 x 1050 px

Collection pages

For the collection featured image, we suggest approximately 1100 x 300-500 px for ShowTime.  For Galleria, we suggest approximately 1400 x 800 px.

Blog posts

The size of images needed for your blog posts is dependent on the settings you use to layout your blog in theme settings.  As a general rule of thumb, an image approximately 1000 x 500 px should work well for ShowTime.  For Galleria, we suggest approximately 1400 x 800 px.

Product photos

For product photos, we recommend that you upload images in as high resolution as possible.  Shopify automatically creates multiple sizes for your product images and the theme uses the correct sizes for different areas of the site.  For best results, all your product photos should be roughly the same dimensions, but if this isn't possible, the theme adjusts them to maintain the integrity of the product grid.  You can read more about this process and how to control it here.