Notice that the brake pedal is hard when the engine is off but softens when you start the car? That is actually normal - and a useful test. Here is what it means.
Normal booster behavior
The power brake booster uses engine vacuum to assist your foot. With the engine off, there is no vacuum assist (after a couple of pumps), so the pedal feels hard and high; start the engine and vacuum returns, softening the pedal; see how boosters assist. This is exactly how it should behave.
The booster test
Mechanics use this: pump the pedal several times with the engine off to bleed off vacuum (pedal goes hard), hold the pedal down, then start the engine. A good booster makes the pedal sink slightly as vacuum assists. No change points to a booster or vacuum problem.
When it is a problem
If the pedal stays hard even with the engine running, the booster is not getting vacuum - a failed booster, a bad check valve, or a leaking vacuum hose; see booster vacuum and hard pedal. That needs fixing - braking takes much more effort.
What to do
If the pedal firms up correctly with the engine, all is well. If it stays hard running, check the booster, check valve, and vacuum source. Keep the friction brakes fresh with quality pads and hardware.