WoW is definitely geared towards team playing, although solo playing is possible, and something that I do myself rather regularly.
However, in terms of getting things done quickly and efficiently, or doing them at a level that the monsters themselves still give decent experience, it's usually way more lucrative to do the "fighting" quests in a group with other players.
As for dungeons, my experience has been with Wailing Caverns (WC) and even though the guys at the entrance are only lvl 16, they have LOTS of hit points and they're elite as well, which means they might kill you in your ghost form! I've never had it happen, but it's Elite monsters that do that... At any rate, in dungeons, parties are pretty much a necessity. I'm sure some lvl 60s could run through the lvl 15-20 WC dungeon, but I'm not talking about that level of player proficiency in this blog.
If you don't have any friends on your Friends list who are interested in partying with you to complete some quests or lay waste to some enemies, here's some more ideas on how to find a party, and perhaps some friends :)
Finding A Party - The Easy Way
[note, this entry pre-dates the LFG tool]
Finding a party can be really easy, or really difficult - depending on what the player population is like on the server at the time and what they're doing at the time.
Sometimes you'll find that there is another player killing the same enemies you are. Perhaps they're doing the same quest you are. Click on them, then right-click on their head-shot in the top left-hand corner of your play window, and Invite them into a group with you.
If they decline, go on about your business. If they accept, you can talk to them using the /p or /party command:
"/p hi there, are you working on the razorback quests?"
You can even refer to a specific quest in your quest log by opening your quest log, then starting the /p message, and where you want the quest info to appear, you shift-click (left click) on the title of the quest in your quest log, and it'll show up in your party chat.
Finding A Party - LFG on Public Chat Lines
A second way you can find a party is by announcing on the LFG (looking for group) channel that you're looking for a party. It's best if you specify what you want to be doing in this party when you announce, so you get responses that are useful right away instead of having to clarify with additional conversation:
/lfg LFG for taking on WC quests. whisper me if you're interested
The translation of this is that you're chatting on the Looking For Group line, saying that you're looking for a group to take on the Wailing Cavern quests in The Barrens, and then you're asking interested parties to respond privately to you instead of spamming everyone listening to the LFG channel itself.
I've never tried this way personally, so I don't know how well it would work. I know I just see others using the LFG line or their guild line to make the requests, I don't know if they actually get a party out of the deal.
Finding A Party - The Meetingstone for Dungeons
At some Inns, you'll find that when you speak to the Innkeeper, there's an option you can choose to have the Innkeeper give you some information on the Dungeons you could take on. If you ask about those dungeons, you'll get an opportunity to be placed on a list of people who are also looking for a party to take on the same dungeon you've chosen.
When another person joins the Meeting Stone party for your dungeon, a party will automatically be created so you can get started with the dungeon exploring. And as a bonus, you don't even have to hang around the Inn to wait - you can head out and start new adventures, and when a party can be created, it will be, and you can go from there!
Solo or Party?
Now, knowing some ways to meet new people and party up for easier quest-completion or waste-laying of enemies, get on out there and get big'n'strong with new friends!