Actually, I wonder if the Space Force will initially be formed from existing units operating in space. For example, we have satellites in orbit for surveillance and communication, as well as the GPS constellation. I wonder if the Space Force would take over management of those assets? Then to put some teeth in space, they could become the primary military interface to the Missile Defense Agency. The MDA, working with the Army, Navy, and Air Force currently has interceptors that work both endo-atmospheric and exo-atmospheric as well as space surveillance radar systems (as does the Air Force).
However, I'm sure the existing branches and intel agencies are not going to be too willing to give up assets, budget, prestige, capability, and control. This will probably be an uphill battle not just with Congress and the leftists 'rats who will oppose anything President Trump does.
The net is - I bet the Air Force is the least excited about this proposal as they stand to lose the most to the new branch.