In the diamond shop there are three types of upgrades that you can purchase; Friendship Items, Decorations and Items.

Friendship items unlock Pokemon which can help you in your game.
For example if you purchase Light Ball you will get Pikachu's help for some league battles. You can also level up each Friendship Item by using support candies. This will make their help more effective.
You get Support Candies as you play through the game such as leveling up Training and food, or beating some leagues. You also get given them on completion of certain achievements in the game.
Decorations are extra items that you can place in your pond to increase the effectiveness of rewards and JP. For example some decorations will increase the amount of JP you get from food, and others will increase the amount of coins you get from events. You purchase these decorations with diamonds which you can either get via playing the game or spending real money on.
There are four main itmes available in the Town Diamond Shop, these are:
Training Soda
Cost: 25 Diamonds
Restores 1 Training Point, you get a set of three of these Sodas.
Really useful if you want another go at Training your Pokemon without waiting for half an hour for a training point.
Power Up
Cost 200 Diamonds
Boost JP gains from Training etc... by 50% for 3 hours
It's a bit a pricey at 200 diamonds and only last for three hours.
LP Restore
Cost 25 Diamonds
Restores your League Play
Essential item if you want to skip to where you left off after your last Magikarp retires. A great time-saving item, so make sure you always have one of these when playing league battles.
Skill Restore
Cost 150 Diamonds
Restores any support Pokemon's skill
Other items purchasable in the item shop are different quantities of coins.