When my visiting physical therapist thought I was strong enough, we did the stairs in and out of the house, always with a cane to support the new hip. Then as I got stronger ( less swelling, too) I did a whole set of stairs, with a cane, but was warned to only do this with someone spotting me.
Then I got to the point this past week she said I could do it on my own, with the cane. Thursday she said I could start to alternate my legs a bit on the stairs. I still use the cane to support the new hip. This sounds a bit terrible, but I used, bad leg goes to hell, to remember that one always goes first, going down, and good leg to heaven, to use that leg for going up. It really helped me remember which leg to use.