Yes, with the stair runner.

Find a railing you love (Pinterest stair railing ideas) and get some quotes.

When you remove the carpet from the stairs you might find plywood subfloor under it. This means you would have to install wood floor treads before doing a runner. So before you embark upon this, check out the price of wood floor treads. It is surprisingly high (speaking from experience here), and you will have to match your existing floor. Unless you bought the house brand new and know there are real wood treads under the carpet already . . .

Agreed the biggest concern is what’s underneath the carpet. And then matching to the existing stain color.
So I’d work on that first.

In terms of railing, posts, balusters I like square (see inspiration below)

Love a runner, and it should match the upstairs. Find a few that you love and come back to us with some options.

If I was going to update, I'd go w/something like this

diff newel posts, more modern spindles, stained treads/painted risers

I love this one too. you could do a runner if you prefer. but changing out the colonial style newels posts and spindles are a must

if all you want to do is the treads, it would look like this

if you want something where you also update the walls surrounding the staircase, you could do this, and iron ballusters

and if you have the budget, then update everything

Thank you everyone for the replies. I too, prefer the square banisters but do like the simple iron rods. I was feeling unsure about going that route with our more traditional crown molding,ect.

We checked underneath the carpet when we had our wood floors restrained about 5 years ago, and unfortunately we will need to install wood treads. That is why it has taken us some time to get around to the project. Luckily, while the floors are original to the house we did restrain them so I think we will be able to match the color.

For the runner, since all of the bedrooms are carpeted and we will likely want to stay neutral in there, will that need to be the carpet on the stairs?

*restained