Repairing these worn spots should be fairly easy assuming the damage is only to the urethane finish and not into the wood itself. This should be as easy as buffing the affected area with a green dish style scrubby to remove loose finish and slightly level the surrounding surface. If the scrubby is not quite aggressive enough, use some very fine 220 grit sand paper gently by hand or with a DA or vibrating style sander. Without seeing the area and damage, it is hard to say how far you need to sand, but most likely do the area where the chairs have worn the finish to slightly into the non affected area. Be absolutely sure to get the same urethane product and finish (satin, semi gloss, or gloss) to do the overcoat. After cleaning all dust etc after sanding, apply a VERY THIN coat of the new finish. Allow this to dry completely, buff sand the area again removing any dust that dried into wet finish, assuring it is extremely smooth to the touch. Repeat this process with at least 3 coats. I recommend using a good grade foam brush or very good Purdy flared bristle brush. If the area is larger, consider a mohair roller. A mohair roller is simply a very low nap roller avail at a good paint specialty store. You may see a slight difference in the new finish to the old surrounding area, simply because of the age of the rest of the room, however that is a small compromise compared to having the damage migrate into the wood which will require a much more extensive repair later. Good luck
Use a hard, inexpensive local wood, without large pores. Hard, since they will get banged around. Pores will allow the wood to trap bacteria, dirt where you don't want it.
Maple is my first choice. Second choice might be birch, sycamore, or alder, if you have a source for them. Poplar is not bad either - cheap, reasonably tough. Cherry would work too. Much depends on what you can find locally. Do NOT go to your local big box store (Home Depot, etc.) to buy hardwood. Hardwood bought there will be quite expensive for what you get.
Woods that I would NOT use are imported exotic woods. These can often cause allergic reactions. Another wood to be avoided for this purpose is walnut, which might cause reactions in some.
Use wood coloring, rather than paint. Paint can chip off, and may not be food safe. And be careful with stains, since many of them will not be food safe. Wood dyes, like aniline dyes, may also be something I would not put in my child's mouth. For a coloring, I would use simple food coloring.
Best Answer
It's hard to beat Polyurethane for durability. I have had good success with both water based and solvent based versions. Get the stuff made for floors, probably two coats would do you. You don't want it too smooth or you'll die when it's wet.
That is, unless you have small children, then I would suggest poured concrete ;)