If you do decide to tile the stairs, make sure they are a matte finish so it is slip resistant.

I would do all the same tile especially if the spaces are fairly small? When one uses more of the same material it tends to give an expansive feel.

Careful that you do not change the rise at all on the stairs, or if you do they all remain the same and withing code. Also what about that railing? You may need a permit and you start on the stairs and the whole enchilda needs to be brought up to current code, and that railing will not fly.

Not sure how to finish the ends.

If they are real wood it would save some money to stain them but you need to try to match the wood look tile. Then tile the landing.

Safety first. I would go with carpet on the stairs only like the blog link Kathryn P posted.