Diamonds are the game's paid currency. They can be earned generously through events such as Monster Riot, Head on Battle, Gym Challenge, by clearing Adventure Maps and completing Achievements. It is not necessary to spend real life money to enjoy this game and be competitive on your server.
If you are a new player it is advised to save your diamonds for 10x Advanced Capsule Pulls with events that give you a rebate on diamonds, but most importantly specific Monster Shards. The best way to earn strong Monsters is by saving 24000+ diamonds and spending them on the Advanced Capsule Pulls where the rebate gives you the Monsters shards you like.
Typically people will aim for Celebi as their first Monster as she is great in early game for PvP and works well in Monster Riot and World Boss events. Personally i would skip Celebi and aim for a weather buff Monster such as Tapu Koko, Tapu Fini, Tapu Lele, Gensect, Kyogre or Groudon. The reason for this is because Accelgor can be used as a replacement for Celebi in Monster Riot and the week day World Boss. Having a weather buff Monster early on will allow you to quickly build a team you can use all the way up to level 95 where Celebi will no longer be used.
When spending diamonds on Capsule Pulls every 25000 diamonds you spend you will get you either a rare or legendary Monster guaranteed with a random personality (exception is Rayquazza after X diamonds). This is a good way to get Monsters but it should never be relied upon or expected to be good since you will pull a completely random Monster with random personality.
What you should/can spend your diamonds for:
VIT: It is recommended to spend up to 2x 50 diamonds on VIT daily to help complete Adventure Maps (you will make your diamonds back if you are getting 4 stars on all maps) not only will this profit you more diamonds quickly it will also help you level your player level up quicker to unlock more ways to increase your overall power.
Fortune Cat: Early in the game Fortune Cat might be usefull and cheap if you need extra gold. But it's not worth to spend money on, as you progress in the game, because it gives you only 20 EXP as a daily mission.
Turnable: Every time there is a turntable event where the only options are diamonds, do it. You will spend 500 diamonds and minimum get 600 diamonds back, leaving you with a profit of 100 diamonds.
Obtaining Legendaries by Adv-Capsule Gift Event: Some players save enough Diamonds to spend on Adv-Capsule Gift Events. They way this works, is that you open a certain amount of 10x Diamond capsule toys, and you get rewarded for doing that. For example, you need 37320 Diamonds (15x 2488) for a Pokémon that costs 160 Shards, but keep in mind that by doing this, you will get Diamond Burning Rewards, Adv-Capsule Rewards, and atleast one Guaranteed Legendary from Capsule toy. So It's way more rewarding.
How else can they be spent?
Diamonds can be used for many things including, Costumes, Furniture, Berries, refreshing Monster personalities for shard summoning, more attempts at events, resetting Adventure Map cooldowns and a bunch of other things.