Item Rewards appear in your Backpack. You will receive a message above your character's head when you complete the quest and receive the item.
Experience Rewards will add to your current Experience points.
Potion Quest Rewards will show up beside your Health and Mana Vials.
Training Point Quest Rewards will add to your current total of Training Points.
If you are on a quest that requires you collect items from creatures during duels, you will receive confirmation messages as to whether you received that item or not after each duel.
The first time you complete a quest within a dungeon you receive 100% of the experience for the quest. The second time you complete the same quest within that same dungeon, you receive 50% experience for that quest. All subsequent times you complete that quest within that dungeon, you do not receive any experience. You will always receive full experience for duels against creatures, just not goals related to the quest.