Soffit, fascia and gutter are usually priced as one line and thought about last. At the eave they are the only thing standing between the roof and the wall.
Soffit
The soffit closes the underside of the eave. Vented soffit also feeds the roof assembly air, which is why painting over the vents or packing insulation against them causes problems further up.
Fascia
The fascia is the vertical board the gutter fixes to, and it is the edge the roof terminates at. It takes the water the roof sheds and hands it to the gutter, so its top edge and its joints are worth getting right.
Gutter
The gutter carries water away from the wall. Its capacity, its fall and the number of outlets decide whether it does that in a storm, and its fixings decide whether it still does it in five years.
At a corner, all three meet a fourth thing: the wall’s own flashing. That is the junction to look at on any building with water in the wall.