You need to mention your Mobile Phone's Brand and model number so that I can suggest you methods to download touch enabled games on your phone, please reply soon.
Well if course. If you have an Android device, just go to the Play Store. If you have an iOS device, just go to the iTunes app Store. Games on smartphones no longer need a keypad or a virtual key board. Maybe you have heard of Temple Run or Angry Birds right? These are cross platform games that only require you to either swipe at the screen or move the smartphone around using motion controls. Both games are free by the way so you should check it out.
There is a big market for touch screen devices. You will find that the best games are made for Apple and Android products, less for Windows phones, but if you make right research in app store, you will find some nice apps.
You should check out Plants Vs. Zombies 2. I think that is one of the games that fits in perfectly with touchscreen gaming and is actually better for it. I actually dislike mere porting of console or flash games to the mobile market myself, but I do understand that some people find that to be a good thing. Still, if you're looking for a good touchscreen game, you can't go wrong with PvZ or PvZ2, especially since they're even free!
No dude, PvZ ain't free on the on the Google Play Store. It's worth slightly above one dollar. Plants vs Zombies 2 is the free game but has different items that you could buy using real cash. You're right when you say that PvZ is great for touch screen phones but this game is not available on low end phones so you must check for compatibility.