What’s a good way to start making money in OpenTTD

I believe the straight forward answer is "no"; as you've discovered the load/unload orders you can give to vehicles currently do not allow you to make them ignore the cargo they have dropped off.

However, as a work-around, have you considered a single combined station instead of a feeder service?

Assuming the bank is close enough to the airport (dependent on your settings, more below), you can Ctrl+Click when placing a loading bay (or equivalent) near the bank, and select an existing near-by station (i.e., your airport) to have the new construction form exactly the same station. Then they share waiting all cargo, ratings, etc - so there is then no need of a feeder service.

The maximum spread of a station is an advanced setting that you can change, and defaults to 12. I'm fairly sure it can be adjusted mid-game, if necessary.

One-way signals reserve a signal block between it and the next signal.

One-way path signals reserve an individual path between it and the next signal. This allows 2 trains to be on the same signal block at the same time, so long as they have paths that don't cross.

From the wiki:

The two new signal types behave a bit differently than standard OpenTTD signal types. The Path signals are red by default, and will only show green as soon as a train can reserve a path to the next safe waiting position on its route. Safe waiting positions are - by definition - in front of signals, depots and track ends. The back of a Path signal is not considered a safe waiting position, and therefore paths are reserved through these signals.

Because the front of every signal is defined as a safe waiting position, you would normally not want to place a signal immediately behind a junction, only in front of a junction. This is because it is only safe for a train to wait in front of a junction. It is not safe for a a train to wait at a signal immediately after a junction before the whole train has cleared the junction, as it would be blocking the junction while waiting, as illustrated in the example below. This is a major advantage against standard OpenTTD signals, where you had to place signals before and after junctions, which caused trains to block junctions while waiting.

For straight track and basic merges/splits, I stick to normal one-way signals. I use path signals for forks that have 4 or more tracks, as this is where they start to benefit.

Path signals also can behave similar to entry/exit signals, although entry/exit signals have uses that path can't fully achieve. For instance, entry/exit signals will cause trains to take longer routes if a path is blocked, while path signals may wait. This can be used to construct balancers and other helpful constructs.