What is this old tile game I inherited?
This sounds like Uno Dominos.

Looking at the Board Game Geek -site for the game (linked above), I found these instructions on how to play.


The design shown in the picture above seems to be a pretty common one. There's another kind of tile design, done by Mattel, that can be seen in some images on the Board Game Geek -site and the instructions linked above. The rules should be the same regardless of what the tiles look like.