For someone that's new to add-ons, try and stick to the smaller ones at first until you get used to what they do and how they work. I started out using one or two addons (the first I ever got was a map/coordinate system called Cartographer - sadly no longer updated), and have slowly installed more as I figured out what I needed.
The addons I use are:
Gatherer
Dominos
SexyMap
Altoholic
X-Perl UnitFrames
Recount
Omen Threat Meter
Deadly Boss Mod
Quartz
Healbot
PallyPower
GATHERER: Great add-on if your toon has mining or herbalism as a profession. This addon works in the background, and tracks mining/herbing locations on the map as you (or your guildmates) gather. The addon creates a mini icon on the map that pop up to let you know there is a spawn point or node of a certain item. It's also smart in that it will only show you herbs when you're on your herbalist (unless you change the settings to show you the nodes at all times). Very low maintenance and excellent if you farm.
DOMINOS: This add-on modifies your action bars. It is very similar to the Bartender add-on mentioned above, but I personally prefer this one. It takes a little playing around to get used to it, but it's easy to learn, and once it's set up you'll wonder how you ever got along without it. Not only can you resize the bars, but you can select how many to show, as well as how many buttons you want on each bar. Just to note: as with any add-on that modifies the original interface, your UI will look a bit screwy if you revert back to the default Blizzard UI.
SEXYMAP: As the name suggests, it makes your minimap sexy. This addon allows you to resize and move the mini map to any part of your screen (which the Blizzard UI doesn't allow), and comes with a lot of fun skins. You can also set it up to show or hide specific buttons (like the world map button, zoom in/out button, add-on buttons).
ALTOHOLIC: I love love love this add-on. I have multiple toons, and this add-on helps me keep track of everything about the toons from their gear to their achievements. I have a lot of bank toons, and rather than logging in and out of each toon to see who has what, this add-on will let you search items across all of yours toons, and let you know if it's in their bag, bank, mailbox, or guild bank. It also has in-game notification about cooldowns for crafting items like transmutes, and gives warnings about mail that may soon expire (never lose mail again!). One thing I dislike - while it will keep track of everything, it will not erase the data from toons you have deleted or guild banks you no longer access. To do this requires you to go into the WTF folder and delete the associated data file (a bit scary sometimes).
X-PERL UNITFRAMES: Fair warning - this add-on is NOT easy to set up. X-Perl, like most other UI addons (Pitbull is another), is a pain to set up and a pain when you need to update or uninstall it. They are horrible come patch day as often times you need to redo a lot of your settings. For new users, you can look at spending a couple of hours learning about the settings and getting things just right. That being said, I still use it because I love how versatile I can make my UI. It allows for full customization, allowing you to decide how you want your character, party, and raid frames to display (changes the looks of the portraits and bars, as well as the sizing of them), and, as a healer, really frees up my screen space for other addons when it comes to raid. You can make it as busy or as minimal as you want, which is great, and you can be very selective as to what information gets displayed. My favourite part - it shows health number AND percentages at the same time for all of my raid members.
RECOUNT: This is really an add-on used for raiding. It tracks everything - damage done, healing given, overhealing given, rage/mana/energy regeneration, deaths, friendly fire, etc.
OMEN THREAT METER: Again, a raid add-on used to make sure you don't pull threat on the tank. Very easy to install and use.
DEADLY BOSS MODS: Either this or BigWigs is essential for any raider. It really helps to remind you on boss fights about special abilities and needed warnings. No raider should be without one.
QUARTZ: This is a cast-bar addon. It will post up a cast bar so you can see the casting time for your spells, but also shows you when you can begin casting your next ability. For chaining heals, I find this one super helpful. It also does quirky things like show a castbar for your hearth stone, crafting time, and mounting.
HEALBOT: One of the many add-ons available to healers. I personally prefer this one as it's an all-in-one addon. This is a click-to-heal add-on, meaning you scroll your mouse over the player's name in the window and click the appropriate mouse button to cast the desired heal. It comes ready out-of-the-box for any healing class (the presets are usually very well don