How do you create a mesh made up of mutliple objects within each other? I am taking part in a 3D printing project, where I am aiming to print a model, made up of three/four cup-shaped objects within each other (like a russian doll essentially).They would be held in place at the top with bridges so they don't move relative to each other.

Does anyone know how I could create a mesh like this in blender?
If I understand correctly what you're trying to make, you can duplicate mesh with Shift+D. Then S to scale inner part to the required size and G to grab and move it.
To connect them you could select 2 edges and press F (only one edge at the time) or use Bridge Edge Loops tool to select two face loops from both meshes and bridge them with faces.
To preview what you're doing use wireframe shading mode (with Z). To have control over meshes, use Mask modifier to toggle their visibility;  Knife tool to cut half of the cup and mirror another part with Mirror modifier; store different meshes on different 3D layers among other tools.