What wood is it?

Has it been varnished previously or just stained raw wood?
If it was simply stained wood I would give it a rub down (along the grain) using 000 grade wire wool dipped in a mixture of methylated spirit and linseed oil. Hard work but gives a beautiful rich result and a little hard paste floor wax to improve the sheen.
If it has old varnish or some mysterious finish in poor condition it will still be hard work, removing the old finish by whatever means. Polyurethane is a good hard finish but don't be tempted to use it anywhere in direct sun. It degrades. After stripping and sanding the old finish off I would use one of the French brand products that are made for high wear areas and stairs called Vitrificateurs. They are extremely tough. Sheen or matt finish and some have a slight tint to enhance the colour. Water based, no stink application by fine roller and brush. I have given up on using any UK products here on wood. Wall emulsions are fine, and cheaper but the Ronseal & Cuprinol products just don't stand up the the weather as well as French products.

The pine will have soaked up the previous treatment and you will need to find a colour quite similar or it will not end as the tin promises. As pine is quite soft, a sand down with 120 grit should clear away old varnish. The chestnut floor is probably a bit like oak and can be cleaned with a wire brush. Try in a corner first to be sure it does not damage it. If the varnish is tough to remove, then 120 grit sander will shift it. Redo the treatment with products as mentioned above. Remember that the pine will soak up any treatment and may end up darker than you expect.

A couple of thoughts...

If you are varnishing the stair treads, go for a polyurethane varnish which claims to be durable.
Also don't go for a gloss finish on the treads (it hurts when your feet slip)

if you and the OH can manage it - do every other step at a time.. the second half tho becomes difficult remembering where to step.