I think astrogation is like flying a jet: most of the time it is about selecting the right option on the autopilot. It is the rare exceptions and overall strategy that require skill.
If the pilot of a ship wants to go from Gateway to a particular habitat over Tirane I would guess the following steps are involved:
The pilot decides on the overall flight, especially what gravity assists and discharges may be needed. The trick is finding a flight that is as fast, cheap, safe and elegant as possible. Since planets are moving, the exact flight time and position might be changed and there might be various delays, the real flight plan is more like a huge bundle of possible flights, with the preferred one at the core. There is good software for this and lots of navigational databases with the orbits of everything major. The software can search out possible flightpaths automatically; most pilots never use anything else. Real pilots of course know how to handle uncertain data, badly charted systems, drive quirks and totally unexpected situations: this is (they say) real astrogation.
Requesting departure from Gateway traffic control, logging a preliminary course out from Earth. Traffic control will give a certain flight corridor to the ship that avoids all the other traffic and objects. The pilot (and likely his organisation) will negotiate timeslots; some spacelines will have preferential treatment, there might be bidding systems for the most desirable slots and corridors.
The ship disconnects from Gateway, and uses its own thrusters and perhaps tugs to move sufficiently away from the safety zone around the beanstalk before engaging the drive. This is basically piloting, knowing how to handle the nontrivial inertia of the ship (especially if it has rotating parts) with a set of thrusters. Good ships of course have software that makes this easy, good pilots know the limits of the software.
The ship begins to move away using the drive. At this point the flight corridor might simply be put into the autopilot until further notice. The pilot will stay on watch for changes (which are likely in a busy place like Earth orbit) and update based on this. Most of this part is merely button-pushing, except for big contingencies like being inspected by the OQC (back to fine grained thruster fiddling again, as well as the prospect of having to redo the flight plan due to the delay).
Once outside Earth traffic control (I would guess about a 100,000 km or more away) things slow down even more. The (best fit) flight plan is run on the autopilot and the crew only needs to check comms and scanners occasionally. The transition to superluminal speed may be impressive for passengers, but it does not make much difference to the crew except that comms become trickier.
If the plan is to do a quick slingshot past Jupiter to get the right velocity vector things speed up again after a few hours. The ship enters Jupiter traffic control space (this is such a busy place that it is needed, although it is not as strict as near Earth), gets clearance for a certain trajectory as well as navigational updates (look out for the sulphur dust trail from that recent eruption of Pillan Patera on Io!). Again the pilot needs to be on watch, but it is likely pretty uneventful.
Once free from Jupiter gravity it is time for the serious part of the trip. Again, most of the job has already been done and it is mostly about loading the right flightpath and cranking up the drive to max. The pilot will likely adjust the ship orientation with the thrusters to ensure that any collision with interstellar dust will make minimal damage (depends a bit on the ship shape and the angle between the realspace velocity and the stutterwarp pseudomovement). Not crucial, but insurance companies and safety inspectors like it.
Once arriving in the Alpha Centauri system things speed up slightly. If necessary more velocity adjustments are done, and since traffic is lighter here there may not be a traffic control near the outer planets. That means the pilot need to broadcast position himself, and make sure he doesn't get too close to any other. The probability of hitting each other is of course essentially 0%, but again safety regulations suggest erring on the safe side - plus, in many systems there might exist trigger-happy police or military ships looking for smugglers, pirates and Kafers.
Approaching Tirane the pilot negotiates a flight corridor with traffic control. This is likely the most annoying part of the trip, since the delays in communications are still long. Depending on traffic, how expected the ship is, relative orbits of the ship and destination etc a corridor is assigned. At any time in a busy system like this there may be several ships "hanging" close to the traffic control zone waiting for their turn.
Again, the autopilot runs the flight corridor until the ship comes close enough that it has turn of the drive, and then thruster autopilot and the real pilot make the final approach (possibly helped by tugs or downloaded commands from the habitat; some habs really like controlling nearby ships directly, something that pilots generally hate).
Finally the ship docks and various connection checks get done, as well as whatever decontamination, documentation and registration the local bureaucracy wants. The pilot also orders the chief engineer to begin stutterwarp discharge (another matter some bossy habitats might want to send a representative to check that it is done properly).
Have I missed anything?
While this kind of routine travel is pretty unglamorous I think there are some potential adventure seeds here. Just imagine the potential if the competitor to a shipping firm could sneak in subtle errors in the optimization of flight plans of their ships, slowing transports? Conversely, what if a brown dwarf was found that helped velocity adjustment rather than length of travel - those coordinates would be a great industrial secret. Political decisions might affect the PCs departure timeslot adversely - if they *really* need to get away, how to do it without breaking the rules? If habitat flight control has access right to send navigational commands, what if somebody hacks it? It might also be fun to pit the frontier pilot used to navigating "by hand" against the intricate rules and etiquette of Core traffic control.
