If you live in an area with a lot of rain or snow, I would go with stone. Agree with the previous post on the lights. I can't see what your front door looks like on the outside. If it's gorgeous, leave it, but from inside, it looks a little boring. Maybe paint it out a color. Also, are your stair treads painted? If so, I would strip and refinish the treads only.

Wood flooring or travertine flooring with very narrow grout lines. Change the door out to a fiberglass stainable door or a fiberglass that is already stained in mahogany. Recess the lights and add a semi-flush mount in the center. Paint the trim white. Are the treads on the stairs painted white? If so, I would change to a dark and install a runner if there is not one.

Thanks for the input. What would you do with the floor?

I meant what would your do with the ceiling?

I can't tell exactly what the ceiling is. Is it stained wood? I would paint it. All of the walls visible in your picture are white. Do you plan on painting the walls?

Found this on Houzz pictures. Are these the tiles which I should look for?

Look at Paris Ceramics website for stone ideas

Great bones, as they say! Color, color…

I would like to paint the walls. Thinking of a soft gray. What other neutral colors should I consider?

With a budget of $10 K a lot can be done here. If this is the main entry area replace the tiles with ceramic or porcelain tiles. Paint the wall and leave the trim white. The lighting fixture can be replaced.
An electrician will give a good estimate as to how much it will cost to replace the existing lighting fixture and instead add one chandelier or lighting fixture. Ceiling color can stay same if you like old world charm. With the color added to the wall the white door and stair banister will stand out and appear attractive. Adding pictures on the either side of blank wall opposite staircase will add visual interest. Instead of small painting add a large painting on the hallway wall. Replace the door mat with a bigger mat. Consistency in color and theme is important to make the space appear bigger and appealing.

I guess what we neglected to ask you is where you are located and what is the overall style of the house? The picture that you showed of floor tile would lead me to believe Southwest?

Would try LED cans in place of the four current fixtures and a decorative chandelier in the center.
Consider greige for paint color. Check for it in Ideabooks here on Houzz.

I would change the arch near the front door to a transom window and I would have a front door with glass in it to let more light into the space. I would change the baseboard to a larger scale and nice trim on the tops of the entrances to the other rooms. I would change the lighting and change the ceiling paint to a lower lustre. I agree that the treads on the stairs should be stripped and stained and add a corresponding wood flooring.

LOVE the whiteness, banister and railing!! I'd get a fancy chandelier w maybe touches of black in it or real crystals in the center ceiling, -that's it. Too much light looks like you're having surgery- I'd get a new fiber glass door w/ beveled (no see thru) glass in the window. Keep the door darkish and the floor vein-y and fascinating. Marble is hard to keep polished, so go and see the amazing new LARGE ceramic tiles that are out there. Makes the room look even bigger. Keep the floor neutral in colors so you won't tire of it. Then you get the beautiful large oil on the largest wall w/ the most perfect colors that you love. Runner up the stairs. maybe taup & black print--I've seen it b/4. Love tour house!