Looking for tips to restore an old chess set

My grandmother recently gave me my grandfather's old chess set (she bought it for him before their wedding, so it's at least 50 years old). It isn't a fancy set, just a plain simple set, but you can imagine it has a lot of sentimental value. Its condition isn't the best, so I would like to make a little restoration on it, mainly to avoid future damage, but also, if possible, to remove small scratches and give back some of its original splendor.

Right now, my only plan is to clean it as much as possible without damaging it, but I think it would need a little more effort. I have never done anything like this, so I would appreciate any comments, suggestions and tips about what to do (preferebly simple, non-damaging techniques). I repeat, my main concern is to avoid future damage.

First, the board. The condition isn't really bad, but I think it could be better. As you can see, the black tiles on one side of the board (upper right on this picture) are damaged, I don't know if this is due to  sunlight or by other causes.

Here, a detail of the damaged section. It looks like it has lost the external layer (Varnish? Wax? Not totally sure of the material).

Also, other parts look like they have been scratched with some kind of instrument, like here:

Lastly, my main concern with the board is its back. It has some scratches that I would like to avoid getting bigger, like these:

Next, the pieces. They are in quite good shape, and my only concern is the usual loss of the external layer, like here:

Here, some more serious damage:

And finally, most of the pieces have lost its base patch, but I think that is easily fixable with a bit of glue and the appropiate material (I don't know what the material is, any ideas here?):

The last thing is not so much of a concern, as it is the box for the pieces. It is in really bad condition, literally breaking apart, but I am still deciding between repairing it, getting a new one or simply throwing it away and keeping the chess set on a table as decoration (and, obviously, to play). This is the box now:

If anybody here has made this kind of work before, I would really appreciate any tips or suggestions of what to do with this set. Also, comments about what to do about the box are welcome.

I wouldn't mess around with the pieces. some beeswax/lemon oil would go a long way with the board. The pieces have seen battle. It's ok to let it show!

if you refelt the pieces, get billiard cloth, not "felt" - 
its thinner and better quality. 

use wood glue. and yes the process is well documented here!

what's your source for buying baize cloth? 

I usually buy billiard cloth off of ebay and have had great results but I'm always looking for something better!

(also I like wood glue... don't want something too too strong because you may have to refelt them again in the future )

IpswichMatt?

Beeswax/lemon oil for the black tiles and the scratches in the upper side or for the cracks in the back?

I really was considering using felt. Thank you very much for telling me about baize cloth, I will look for it.

Also, what would be the differences between wood glue and contact cement, appart from the time to dry, the hold capacity and the removal difficulty? Is any of them difficult to get or does any of them require a special procedure to apply? In case of doubt, I would like to pick the "safest" option to avoid possible damage to the pieces. Thank you very much!

When you do pieces, do you first cut circles of cloth and then glue them to the pieces, or do you first glue and then cut?

DO NOT use a blow torch

Glue, let the glue try and then cut. There’s a video on Alan Dewey’s chessspy website where he demonstrates the technique 

Exactly right

Unbelievably you can use black spray paint for the black pieces, the stuff made for cars. Obviously you’d need to carefully mask off the the rest of the piece.

Whether you want to do this is another question!

I think I can live with the pieces as they are, only changing the baize. I am more concerned about the cracks on the bottom of the board getting bigger.

Also, I would like to fix the small problems of the upper side of the board, if the techniques are safe. What about the beeswax dajamps suggested?

Also also, what solvent should I use to remove the current remains of baize? Alcohol? Acetone?

Thank you very much for your comments, they are really helpful 

I just scraped off the old felt. It usually gave way with little resistance. 

one tip if the old felt doesn't give way easily - spray goo gone or goof off on it first. Saturate it, then it will come off and leave basically no residue!!

Are these cracks? You're talking about the underside of the board, right? It looks more like the veneer is coming off. If you want to stabilize it you could try minimum viscosity superglue on the underside, it should wick under the veneer and strengthen it. Try a tiny bit first to ensure you're happy.

You'll want some acetone handy too if you're using superglue, it's quite annoying if you get it on your skin

Yes, it is the veneer (didn't know the word) coming off that concerns me. You can literally peel it with your hands!

Do you mean lifting it with care and applying the glue in-between or just applying it on the outer side and letting it infiltrate?

Thank you for your comments, they are really helpful, and my board will thank them for sure.

I meant just let it infiltrate. I used the word wick because that’s how wax goes up the wick on a candle. It’s actually called capillary action, since low viscosity super glue has very low surface tension it’s very good at infiltrating. 

Try just a drop or two at first and see if you’re happy. It is possible to wipe away the excess with acetone, even after the glue sets.

You’ll probably want some “super glue precision tips” too, search on eBay for those and you’ll see what they are.