Questing is simply doing quests that you are given and after finishing them you get experience and rewards. Additionally you get the experience when you have to kill different kinds of mobs. Then you when you level up you get more quests that require higher levels and that is what you do up to max level.
Grinding is when you find a good spot in the area where you can easily kill the enemy mobs and they give you a good amount of experience and loot. You want the mobs to spawn fast so you have always some to fight with. Grinding is not the best solution for every class. I started with a protection paladin and those are probably the worst to grind with since it would take a long time. You want to have high attack and finish your enemies quickly. Your grinding area should be slightly lower than your character level so you don't have any big problems with surviving. You should also have lots of food and water or other things that will replenish your health and mana if you need it. Grinding can be very effective if you have mage for example, when you can kill enemies fast and replenish with eating and drinking.
So why do I think grinding is not as good as questing? First of all, this is completely personal decision and you can prefer grinding if you think it is more efficient and fun. For me grinding is not as much fun as questing. Also with questing you will get more good items than with grinding. Yes you also get green and better items if you grind, but with each completion of a quest you usually have an option to decide what item to get. This is very nice when you need a certain piece. Personally when you do quests you also get to discover more of the map and see different areas which is more fun. For grinding you don't really need a lot of guidance if you know where to stay and kill. With quests that can be different.
A good horde or alliance leveling guide can be or real help. There are special add-ons available that will tell you exactly what quests to do first and where to go to finish them, what skills to learn and what items to buy for best leveling speed.
Click here to find out what are the fastest horde or alliance leveling guides