There's more to it than picking screen resolutions.
Lets look at this problem a little deeper.
Game Media
If you change the screen resolution with
Set Display Mode, you will need to reload all of the media your game is using. It has the same effect as losing the Screen Device. All images, sprites, objects loaded will be deleted and will need to be reloaded if you want to continue to use them. This can be very problematic if you are using an in-game menu.
Screen Aspect Ratio
If you change the resolution, the screen can stretch different ways since most resolutions have different screen shapes and form factors. You need to correct the Aspect Ratio of the screen otherwise the game will look stretched.
Secondary Cameras
If you use additional cameras for effects like reflective water, HDR, Bloom, etc. In most cases those cameras will need their Aspect Ratio to match the Aspect Ratio of the main camera. Otherwise you will see errors, and tear your hair out wondering why.
Desktop Resolution
To a lesser importance, if you are trying to establish a best possible resolution and there is no existing setting or default. Don't set the resolution higher than the desktop resolution. It's a detail that's easy to overlook.
Resolution Reversion
A popular mechanism these days (for good reason) is to provide a timer countdown after changing resolutions in case an undetectable problem occurs and the player can't see the screen. The idea is that the game will swap back to the old resolution when the timer runs out.
A few of these things are easier said than done. However, keeping these points in mind will help you write better software.
