First, make sure you're in the Play Money lobby, where all games are Play Money ONLY.
If your balance on the opening screen shows a real-money currency, go to 'Account' and toggle the option to 'Play Money'.
You can reach all game lobbies from the Play Money Poker screen (Cash Games, Sit & Go, or Tournaments) by tapping on the respective menu at the header bar. Every game lobby has a similar structure. Either you'll see games listed right away, or you have the option to scroll through formats and choose the variant you're interested in.
Here, you'll be taken right away to the games' list.
The filters are a great way to customize your search.
Here, you can choose from different formats (i.e., Hold'em, Omaha) and various variants (Single-Table, Multi-Table, Heads Up, etc.).
To search for a particular tournament (with tournament name or ID), use the magnifying glass icon in the Poker lobby.