One of our favorite beginner phone or tablet user game. This really isn't a game, but teaches your child about touching your phone and getting a reaction. My boys played with this till they were at least 3. Touch the screen to draw or make shapes.
This is one of our favorite coloring apps. It is fun because your child can pick a color and just touch a section to fill it with color. No coloring in the lines just easy fun with lots of diff sheets. Zane likes to color the whole screen one color.
This has always been one of our favorite apps. I first saw it on an iPad and I was so excited to find it on Android. The boys love it. They slowly learned to not touch the bees and they learned about games that you had lives in.
Once you have introduced your child to LEGO DUPLO Zoo you may want to see how they do with LEGO APP4+. Zane actually got this app way before the age of 4 and both boys have loved playing it over and over.
I saved the best for last. Kid Mode You can set up each of your children to be able to log into their own version of Kid Mode. Kid Mode collects dozens of games and videos from online that are age appropriate for each child. Plus it Locks!