I know that air goes into the compressor as the first step of the journey
From the way you mention this phrase it sounds like you have a slight misunderstanding of how a jet engine works.
As your confusion illustrates. Air does not just "flow" unless there is a pressure difference. A pressure difference can be created by several means: atmospheric heating and cooling (which is how we get wind), airfoil moving through air (fans), impellers accelerating air around a circular path (blowers) etc.
Air does not go into the compressor. The compressor pushes air through the engine.
In a modern jet engine the compressor is designed like a fan. Thus the action that causes the air to flow through the compressor is the same as a table or ceiling fan: the airfoil of the compressor blade moving through air. Just like a regular fan you find in most homes, this causes the airfoil to create a high pressure region behind it causing the air to be blown backwards and a low pressure region in front of it causing air to be sucked into the engine.
In older jet engines from the 1940s and 1950s the compressor works kind of like a leaf blower or a car turbocharger compressor with impellers that push air sideways due to centrifugal force. Again, this causes a high pressure zone behind the impeller and the exit of the air inside the impeller causes a low pressure zone in front of it sucking air inside the compressor.
As in any application using fans, blowers or compressors such as a table fan or hair dryer or a turbocharger it is the motion of the compressor that causes air to move in a jet engine.
But unlike a table fan or hair dryer it is the hot jet exhaust that powers the compressor which is the same as how turbochargers work in cars. In fact both aircraft jet engines and car turbochargers are called the same thing: turbines.
How a jet engine is started depends on the design of the engine. Some use an electric motor, some use hydraulics, some use a piston engine and some use a separate "starter" blower to initially blow air into the engine to start the compressor spinning. There are even jet engines designed to start spinning the compressor by firing a shotgun shell inside the engine: these are called catridge start (google "catridge start jet engine" to find interesting youtube videos). You can read more about the various ways to start a jet engine (technically called "gas turbines") on wikipedia: https://en.wikipedia.org/wiki/Aircraft_engine_starting#Gas_turbine_engines