In my experience, it is very easy to solo. In fact, you can solo a major portion of the quests (which are where the real exp is at). You have to be careful about taking out mobs 1-2 at a time, but it can be done. A few quests are meant to be done by small groups (can think of a few "kill named" quests that fit the description), say 2-3 people. If you have a high enough level advantage you may be able to complete even those by yourself.
HOWEVER, some of the quests can NOT be done by yourself. I'm speaking of the "elite" quests. Typically, these bring you to instanced dungeons. These quests are meant to be done by parties, preferably full parties. The best loot and exp is there, and so is the greatest risk and challenge.
That said, the rest of the game can be done by yourself (more or less). I did a lot of soloing. Sometimes it helps if you temporarily group with another person doing the same quest, then go your seperate ways once you've done the job. Other times you really need the backup, like when you're trying to tackle a quest above your level, or the aforementioned elite quests.
Someone mentioned that you can't simply farm low-level monsters to level up. This is both true and untrue. You get zero experience for fighting monsters six levels below your own; the exp harvest on things three levels below your own is paltry anyway. I found that an area with things between 1 and 3 levels below you can be farmed pretty successfully. A warning, however: a lot of areas have roaming monsters who can prove a nasty surprise to someone who's just farming!
To whoever said that one of WoW's virtues is that you can play for 30 minutes at a time and accomplish something significant: right on.