I only have one in use, the default Excalibur. Either I don't have another slot or I don't see where to toggle.
Edit: to be clear, I'm going to Arsenal then pressing X (equip) on the Warframe (which is Excalibur, the only warframe I had prior to claiming 2 in the foundry). And the other warframes that show there require purchase, including the 2 I built and claimed (ember and rhino).
You make 3 parts of the warframe and then you need the blueprint for the completed warframe. So you will need 4 blueprints in total before it completed. Also you have to wait for them all, so best to do the basic 3 at the same time if possible.
I didn't buy anything, I acquired the blueprints as item drops from bosses.
You get the component blueprint but you need to buy the final one from the shop to built the frame.