What sort of exporting format should I be using? Or will I have to re-assign all of the textures again? Thanks.
As atek said, just press A to select all of the objects in your blender scene that you want to be copied over. press Ctrl C to copy and then just simply open up a new blender file and press Ctrl V to paste those objects into your new scene. this will bring all of your textures with it.
What Sarcha & atek have proposed here will indeed work however, it is not really the best way to go about it as it would be handy to have the particular object(s) available for use over multiple files and projects. The 'proscribed ' way to do this is using file>append. In the scene that you want to move the object from select the object and make sure that you have it named in the outliner. (If you cannot find it in the outliner due to many unlabelled objects select it in the 3d viewport and then hover over the properties window and hit '.' (period key on num pad.))
Save the scene and close it. (If you have multiple objects that you want to move then highlight them all and make them into a group using cntrl+G.) Name the group.
Now open the scene that you want to place the object in to and go file>append> find your blend file > meshes, (find your labeled mesh,) and click on append from library. (In the case where you have a group labeled go to groups instead of meshes.) all attributes including texture will then come into the new scene.