I've been doing some digging into what Space Force might look like. Here are my thoughts:
Outer Space
Outer Space begins at 100k or 62 miles. This is the lowest altitude at which orbit is achieved. This is known as the Kármán line (source: https://en.wikipedia.org/wiki/Outer_space). The Outer Space Treaty (established in 1967) represents the basic legal framework of international space law. Among its principles, it bars States party to the treaty from placing weapons of mass destruction in Earth orbit, installing them on the Moon or any other celestial body, or otherwise stationing them in outer space. It exclusively limits the use of the Moon and other celestial bodies to peaceful purposes and expressly prohibits their use for testing weapons of any kind, conducting military manoeuvres, or establishing military bases, installations, and fortifications (Article IV). However, the treaty does not prohibit the placement of conventional weapons in orbit and thus some highly destructive attack strategies such as kinetic bombardment are still potentially allowable (satellite killers, for instance, have been tested). Outer space is considered to be above national boundaries and hence open to all.
For an airplane flying higher and higher, the increasingly thin air provides less and less lift, requiring increasingly higher speed to create enough lift to hold the airplane up. It eventually reaches an altitude where it must fly so fast to generate lift that it reaches orbital velocity. The Kármán line is therefore the highest altitude at which orbital speed provides sufficient aerodynamic lift to fly in a straight line that doesn't follow the curvature of the Earth's surface.
FYI: the United States designates people who travel above an altitude of 50 miles (80 km) as astronauts.
Mesosphere (also known as 'near space')
(see: https://en.wikipedia.org/wiki/Mesosphere) This is the area of space which is typically above the height at which commercial airliners fly, but below the height at which satellites operate. Near space was first explored in the 1930s. Notable people who flew in near space were Jean Piccard and his wife Jeannette, on the nearcraft The Century of Progress (see: http://www.century-of-flight.net/new%20site/balloons/stratosphere.htm). Later exploration was mainly carried out by unmanned craft.
The exact upper and lower boundaries of the mesosphere vary with latitude and with season (higher in winter and at the tropics, lower in summer and at the poles), but the lower boundary is usually located at heights from 50 to 65k (164,000 to 213,000 ft; 31 to 40 mi) above the Earth's surface and the upper boundary (mesopause) is usually around 85 to 100k (53 to 62 mi).
For military purposes, the mesosphere is the area of the earth's atmosphere which is of most interest, and this area is where test vehicles are believed to be under development. This is the area of space in which the U2 spy plane operates.
Low Earth Orbit (LEO)
Low earth orbit is between 160k or 90 mi and 2,000k or 1,200 mi.
Most satellites and all manned space stations are stationed in Leo.
FYI: the re-entry interface is at 122k or 76 miles.
Geostationary orbit is 35,800k or 22,236 mi above the equator. Geo orbit is a very special belt and the speed of a vehicle which is stationed in this orbit equates to an orbital velocity of 3.07 km/s (1.91 mi/s) and an orbital period of 1,436 minutes. This ensures that the satellite will match the Earth's rotational period and has a stationary footprint on the ground. All geostationary satellites have to be located on this ring. There are only a limited number of slots available in this orbit for new satellites.
Space Force
Based on the above, I'd say that the first objective of any Space Force is to operate at altitudes of between 40 - 62mi. Technically speaking, anyone flying a craft above 50mi is designated an astronaut - so on this basis, Space Force doesn't actually have to go into space. The second objective will be to operate above 62mi, as this area of space is free from national boundaries, and below 76mi, to avoid the complications of re-entry. Any (conventional) craft operating in this area of space must be travelling pretty fast.