For something like a homepage slider, we recommend staying in the range of 1440 x 500. Not too tall and not excessively wide.
For a full width and full height images like a background for a page, we recommend staying around 1920 x 1080. This means you are natively at full resolution on an average sized modern monitor.
We always recommend running your images through a compression service such as TinyPNG.com which is a free service that will cut the file size of your images drastically without losing quality. This means your images load faster.