What, in your opinion, is the best way to setup a network of WP sites with a shared login?
- I want users to stay logged in to their account when they switch sites
- I want this to be used with different domains
What you're asking for is very easy with this plugin: http://premium.wpmudev.org/project/domain-mapping/ It does cross-domain cookie syncing with all of your sites on one Multisite installation.
I've used this a couple times and it works dandy.
As @Rarst mentioned above, is anything wrong with setting up WordPress as a Multisite see the Codex Here.