The resolution of your screen has nothing to do with the size of images you can create.
I can create a 1200x1600 image on a 600x800 screen. It just means I'll have to do some zooming and panning and scrolling to see it at actual size.
It would help to understand how you plan on accommodating different screens. Assuming you want to make a full-screen game, there are a few scenarios you could consider:
Design your imagery at a specific size, then stretch-to-fit.
- Pros: One set of assets. One 'playing field'. If you design them fairly large to begin with, they should render fairly well even when stretched.
- Cons: You need to account for aspect ratio differences (perhaps letterbox as needed). Shrinking down too far or up too far could produce a reduction in quality that is unsatisfactory.
Design your imagery at a specific size, but then crop/expand the playing field rather than stretching.
- Pros: Assets are shown at 'actual size'. There is no aspect ratio issues.
- Cons: Depending on the design game, this may or may not work.
Design to specific screen resolutions.
- Pros: It will look 'perfect' on each screen you design for.
- Cons: It may require creating several versions of each asset.
Build the game using vector-based assets.
- Pros: Vectors can scale up and down without a loss in quality.
- Cons: Not all platforms/gaming dev platforms support them.