...ummmm
Or you could just play in any resolution and scale the screenshots to a suitable web size.
All image software tends to offer standard screen resolutions for scaling, and using Irfanview or similar means you can just point it at a folder and let it convert everything.
BTW. Filesize is only half the issue. You can easily post a massive JPG with a tiny filesize if you enable the useful optimisations. Just as easily you can make a low-resolution JPG with a huge filesize if you disable all the optimisations.
The issue with large images is that different site software and browser configs, may or may not scale them to fit the viewable area, so often people can only see the top left corner of the image.
Use RIOT in full auto mode to make sure your images are sensible for web pages (eg. fast). It is also available as a plugin for Gimp and Irfanview.
http://luci.criosweb.ro/riot/
Note: If you upload large pictures here instead of using off-site links, then a thumbnail is presented so it fits no matter how big the images.