For the state you are stationed in, you will have to check on the reciprocity of it with the state your concealed handgun licensed was issued with. Unlike your driver's license, it is not automatic. There are many websites that keep current of such agreements, such as:
https://www.usacarry.com/concealed_carry_permit_reciprocity_maps.html
As for carrying on base, it is dependent on the base commander's policy. Federal law does not prohibit having a personally-owned firearm on military bases, only in Federal facilities (18 U.S. Code § 930), but some service policies (such as AR 190-14) restrict, regulate and/or limit it. Know before you go.
But as stated, purchasing firearms in your state is allowed with your PCS orders.