A simple way to set your screen resolution and color depth for most Live Linux distributions is via the use of the vga=parameter boot option. For example at boot you might type: vga=795. This would set your Linux system to boot with a screen resolution to 24bit 1280X1024.
Here are some more examples of common vga boot values, related to depth and screen resolution;
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Linux VGA Modes Resolution and Color Depth Chart
Note: That's the Old School method...
Here's How To Set Grub VGA Modes and Resolution today;
Grub VGA Modes Autodetect Resolutions
With recent GRUB configurations, modes can be set by simply adding the resolution and color depth to your grub.cfg file. For example;
Syntax is width x height x depth.
Note that depth is optional, and you can use auto to try the next available resolution until one that works is found.
Multiple resolutions can also be provided. Then, during boot GRUB will autodetect which modes will work best with the hardware on the system it is booting from. This allows support for multiple resolution + color depth options for varying systems. Which is exactly what you want when running from a Linux Live USB. See the example below;
In the example above, Grub will first attempt to use a 1920x1080 resolution. If that fails, it will move to 1024x768x32. And if that fails, it will automatically continue to try the next available resolution, until a working resolution for the hardware it is booting from is found.
Note that set GFXPAYLOAD=keep preserves the detected GFXMODE.