I would use a type of cardboard called "illustration board" that can be found at arts & crafts stores. I used a steel ruler, cutting mat and craft knife to cut it in the past... it takes several passes per cut.
I would print up the tiles on 110 lb. Cardstock, cut that out and glue it to
The illustration board pieces.
Same goes for counters.
I used what was supposed to be the covers of a scrap book at Hobby Lobby. It's very hard but not too think of cardboard (feels more like chit board). I agree with above mentioned cardstock option for the printing. It works great. Much better than laminating.