In general, are probationary firefighters (those who have completed their training and are working as fulltime firefighters) generally assigned to engine companies first?
As with all things firefighting, it’s highly dependent on the particular department.
In general, though, usually an engine or a ladder. In departments that also run the ambulance, they are very likely to be assigned to an ambulance first.
Usually the new person is assigned to wherever there is a vacant spot. Someone else retired, promoted out, or resigned and now there’s an unfilled position in a company on one group. If it was a spot that was highly desired, someone else is going to bid into that opening, leaving a less desirable spot now open for the new firefighter. If another person wants the now new spot, they bid to it, etc. and repeat ad nauseam. This is why the ambulance is the first spot - its often the least desired and as soon as something opens up elsewhere, they bounce.
In departments with specialty units, such as a heavy rescue, fire boat, or an airport firefighting station, you usually need some seniority to bid into those spots. Occasionally no one wants that spot and the new firefighter gets it (that’s usually because there’s a crappy officer on that spot).