First things first, until you get to level 70, you always, always want to be doing quests. To paraphrase my favorite David Mamet movie, A, B, Q-- A always, B be, Q questing. Questing is for closers, if you will-- especially at the lower levels, it's the absolute quickest way to earn both experience and money. Later in game, there will be spots where grinding will earn you more money than questing, but starting out, you want to do as many quests as you find, because the early quests will give you everything you need-- money, XP, and even great gear. You can have a limit of 25 quests in your quest log, and whenever possible, you should have all 25 going. Whenever you show up to a new area, walk all over the main town, and look for those little exclamation points everywhere. Even if a quest is higher or lower than your level, take it-- you'll be able to watch it drop to your level as you work your way up, and if you're out and about when it's fallen down to yellow, you'll be all set to do it.
Once you've got all the quests you can carry, the next step is to group them up as much as possible. On your first time through the game, it might be a little hard to figure out where various quests take you (though that info is usually readily available on databases like Thottbot and Wowhead), but even if you don't want to spoil things by looking them up out of game, rereading the quest text will usually tell you exactly where you'll need to go. If a number of quests mention "going north" or a certain area on the map, group those together and do them all at once. Lots of early zones are pretty spread out (Duskwood, for instance, is notoriously bad, with the town at one end of the zone, and most of the quests at the other), so stacking quests is extremely necessary to save time and aggravation ("I just went to that stupid cave! And now I have to go back?!").
Same thing with group quests and dungeon quests-- wait on doing 5man quests until you've got a few of them in the same zone, and then do them all together with a group. Some quest chains will require you to go into dungeons multiple times, but for most dungeons, quests can be grouped up together, and done all at one time on one run through. Always group up like quests, and then knock them out one after another.
Fortunately, there are addons out there designed exactly to help you do this, which leads me to MonkeyQuest. Go-- download it, and install it right now, because even if you've never heard of it, MonkeyQuest is one of those addons that you may have never known existed, but can't live without after you've installed it. It not only will give you an always-open window to sort and view your quests at a click, but it'll also embed itself in tooltips, and nothing is more helpful than seeing a mob on screen and being told something you would have never known otherwise: that it's a quest target. MonkeyQuest will also do more great things like tell you the level numbers of a quest (so you don't have to decode those colors), and it will now even track party members, so groups can better track who's doing what quests. Can't recommend it enough for anyone questing on a regular basis, which should be everyone playing the game.
Finally, the last questing tip for beginners has nothing to do with quests at all-- it has to do with all of that traveling time in between. Just like you should always be questing, you should always be earning, whenever possible. When traveling back and forth from your quest area to the turn-in, grab a few mobs on the way and take them down. There are tons of things to kill for experience in WoW, and if your goal is to level up, then you have to look at how fast you're constantly bringing in the XP-- any downtime will slow you down. After 40, when you get a mount, things can change, because it takes time to mount and unmount every time. But before 40, you should be fighting even when traveling-- demolish everything in your path, and all that XP will add up.
Quest wrangling can be an art in and of itself-- the idea is to get the most done in the shortest amount of time, and while knowing where everything is already can help (most players, I'd say, are able to level alts to 70 much faster than the first time their mains went through, if only because they know where the monsters they need to kill are), so can organization and planning when it comes time to tackle and clear out that full quest log.