Each set of dice and each theme unlocks when you have 5 of its shards. Shards come from chests: any chest can contain a shard, and a Legendary Chest always has one. You get chests by winning Gameroom matches, from Challenge steps and Dice Darts, and as leaderboard rewards.
You do not have to wait for all 5. In the Shop, tap a dice set or theme and unlock it with rubies. The price depends on its rarity, and it goes down for every shard you already have.
A few dice and themes come only from special offers in the Shop or from tasks.