There is an item shop next to your team's shrine - left click to open it.
When starting the game, it's generally good to purchase some health regeneration - you can buy a permanent item like Regrowth Pendant or Rejuvenation Bead from the Defense/Health Regeneration tab, or Health Potions from the consumable tab.
There are 3 lanes where the primary action for Summoner's Rift takes place - one that runs along the left and top sides; one that runs along the bottom and right sides; and one that goes right down the middle. If you're confused about where to go, wait for minions to spawn from one of your inhibitors and follow the minions.
Try to split your team evenly among the 3 lanes.
To learn an ability, click the "+" above it.
To gain gold from killing minions, you have to cause the killing blow, or 'last hit' the minion.
Destroying enemy turrets gives your entire team gold and experience, even if dead, so try to help your team destroy turrets as quickly as you can.
Destroying an enemy inhibitor temporarily causes your minions to be stronger, but the enemy's inhibitor will respawn after a few minutes.
Turrets are very powerful - make sure you have plenty of support when approaching an enemy turret. Conversely, when being attacked, retreat to a turret for more support.
Attacking an enemy champion will cause all nearby units to attack you, including turrets!
Even enemy minions deal a lot of damage early in the game, so try to use your own minions to shield you from taking damage.
You can always go home if you want to - press 7 to teleport to your team's Spawning Pool. It will take a few seconds to take effect, and it can be interrupted by enemies!
Every champion has a passive ability that you always have, even if you didn't level it up. You can view its details by pressing 'C'.
Ability power increases the damage and healing your abilities cause; it does not benefit attacks or buffs. Some abilities benefit from other stats like Attack Damage, but only if specified in the tooltip.
Feel free to ask any question you like, and if appropriate, I will add it to the list.
Thanks and happy LOLing!