you can import a project into another project. this way all the assets (images, footage, solids, audio) will come with it.
i would select the comp (in the project window) that you want to move to another project, then choose file>reduce project. this will remove all items that are not needed for that comp (it will keep any nested comps). then save that as a new project.
open the project that you want to bring that comp into and choose file>import>file and select the project you just saved. it will then bring in that comp and all associated assets.