Diamonds are a key currency whose uses become apparent immediately - they are primarily for summoning dragons. Most uses of diamonds are for obtaining more resources, primarily:
You will see your diamond supply ebb and flow a lot. The game is very generous with diamonds. Below is a table that illustrates the minimum amount of diamonds a single user can expect to get over a 1-month period of 28 days:
The game offers huge diamond rewards for even minimal effort! This is the absolute minimum you can expect every month and is not including:
In reality, you will likely get a lot more.
Primarily you will wish to spend your diamonds on Advanced Summons. Summoning is necessary, not only for seeking new and strong dragons, but also for getting fodder dragons to rank up your team. See the page on Summoning for more information.
Summoning is your main priority early game.
The Witch's Cauldron sells items such as Relationship Points and Skill Slimes - these are also very much worth the diamond investment! Runes, however, are not worth buying with diamonds, as they are very expensive, and you have no guarantee that they will be worth the cost after enhancement. See the page on Witch's Cauldron for more information about the items in the Witch's Cauldron.
Finally, you can also spend diamonds on Wings and gold. I highly recommend buying Wings with diamonds when you need them, and picking them up whenever they appear in the Witch's Cauldron on discount, too. You will always need wings, so it is worth it to pick them up whenever you can get them cheaper.
I do not suggest buying gold with diamonds, as gold is a resource that is always needed in high quantities, but is easily spent. While the other purchases mentioned here are an investment - such as summoning dragons to make other dragons stronger, or spending on Wings to gather resources - gold can be used up in the blink of an eye. However, if it appears on discount in the Witch's Cauldron, you may wish to purchase it if you have no other pressing demands for your diamond supply.