Both vary a lot based on what effect you’re trying to achieve…
For background images… probably as wide as you can get while keeping the file size reasonable (say maybe under 150k or so? a service like https://tinypng.com/ can help reduce filesize). Probably in the range of 1440px - 1920px wide…
For category logos/images I personally like squares, around 150px or so… but if you’re customizing the layout otherwise really any size/orientation can work well.