How can I see if my flights are running?

I have set up base, added flights to a plane, but no way to see if they’re taking off or how many passengers aboard / profit etc. Am I missing a step?

there are 2 ways.

1) in the inventory.

choose a destination, scroll down and you have a list of your flight nos.

at the right hand side of the list is a small triangular arrow you can click.

2) in the fleet management you cann choose each of your aircraft.

on top there is a external map, very nice work based on google maps:

https://www.asroutemap.info/rm.asp

Did you activate the flight plans?