Krdytkyu

When selecting the background for my Sorcerer, I chose Outlander (PHB, p. 136-137). This background gives a "staff" for starting equipment (among other things).

Does the staff act like a quarterstaff or an improvised weapon if used to attack?

This would definitely be a question for your DM, but I believe most people agree that it can be used as a quarterstaff.

There are three things in the equipment list that are within the price range of background kit and could conceivably be described as "staff" - the Staff arcane focus, the Wooden Staff druidic focus, and the quarterstaff itself. For the focuses, it appears that there's a decent amount of support for the idea that one of them can be used as a quarterstaff.

Can the arcane/druidic focus staff double as quarterstaff?

Of course, if it's the quarterstaff, you're good to go.

As for which it is, it's unclear. The most obvious from a pure RAW standpoint is the one simply called "staff" (the arcane focus). The second most obvious is the druidic focus. Those raise odd questions, though, in the case of an Outlander who is neither a druid nor an arcane caster. Perhaps the simplest answer would be for the player to pick one of the three. The most expensive of the lot is 5gp - little enough that it shouldn't be a big issue one way or the other.