Part 87, more properly 47 CFR § 87 defines many different types of stations that use the FCC's "Aviation Services" (the legal use of radios on aviation bands in the US). See 47 CFR § 87.5 - Definitions. Here are some examples:
Aeronautical advisory station (unicom). An aeronautical station used
for advisory and civil defense communications primarily with private
Aeronautical enroute station. An aeronautical station which
communicates with aircraft stations in flight status or with other
aeronautical enroute stations.
Aeronautical multicom station. An aeronautical station used to provide
communications to conduct the activities being performed by, or
directed from, private aircraft.
Aeronautical search and rescue station. An aeronautical station for
communication with aircraft and other aeronautical search and rescue
stations pertaining to search and rescue activities with aircraft.
[I'll skip a few...]
Aircraft station. A mobile station in the aeronautical mobile service
other than a survival craft station, located on board an aircraft.
[There are several more.]
So before you apply, you'll need to know exactly what kind of station license you're applying for.
Section 87.18 says, "An aircraft station is licensed by rule and does not need an individual license issued by the FCC if the aircraft station is not required by statute, treaty, or agreement to which the United States is signatory to carry a radio, and the aircraft station does not make international flights or communications."
So you use your shiny new radio in an aircraft without a license, as long as you don't plan to cross any borders. However, if you don't have a license and you want to test your radio, then you should probably be testing it from inside an aircraft. (Take my advice with a healthy dose of skepticism, because I'm not a lawyer.)
On the other hand, Section 87.47 says, "A person may apply for a portable aircraft radio station license if the need exists to operate the same station on more than one U.S. aircraft." So if you want to use your shiny new radio in more than one aircraft, then you may apply for a portable aircraft radio station license.
The way to apply for any of the Aviation Services licenses (besides the basic aircraft station license for which no license is required) is to register with the FCC's Universal Licensing System (ULS), and then file a license application electronically there, which will be the equivalent of the old Form 601. Depending on how you applied for your restricted radiotelephone license, you may already have the FRN (Federal Registration Number) to use with the ULS.