In my experience, the best class to play as a casual player is one that is easily soloable, with little downtime, but also able to find groups quickly when necessary, particularly at max level. Following is how I rate each class according to those criteria.
Rogue: Sneaking past mobs you don't want to mess with and stunning your enemies so they can't fight back make the Rogue a fun character to solo. They are extremely gear dependent, however, and self-healing is limited to profession-related skills. While rogues make a fight go quickly, there are many of them and you'll have a hard time finding groups while leveling and in the endgame. (Edited to add note: Many readers have pointed out that there aren't as many rogues as there used to be. Perhaps there are more leveling on PvP servers. Ganking me. Like just now. Regardless, there are many DPS classes vying for group spots and rogues still have a harder time finding groups than all tanks, healers and DPS classes who can also heal as necessary.)
Warlock: Warlocks are extremely easy to solo. They have pets, they have self-heals and they have nasty damage. Their spells make downtime pretty non-existent and they can often solo group quests. A skilled Warlock can find groups at max level, but they are not in high demand at this time.
Warrior: Warriors have become much more fun to solo, but are still hampered by a lack of class specific self-heals and therefore require some downtime. If you do make it to the endgame, however, you will find that tanks are in high demand and that a skilled DPS Warrior can also get work.
Shaman: Leveling a Shaman is not difficult. You can blast, melee and heal. If you want to hang out with the big boys in the endgame, however, expect to keep one of your specs Resto, because non-healing Shamans just aren't in very high demand.
Priest: It takes a while to pick up speed when leveling a Priest without a lot of grouping. Shadow Priest soloing is the fastest and gets much easier when you get Shadowform. If you take one to the endgame, you'll be the belle of the ball. They'll want you for your heals and Shadow Priests can even get spots in raids these days.
Mage: Mages are definitely soloable and are very fun to play, but they tend to have a lot of down time for healing and mana regeneration. With no non-profession-related self-heals and nothing but comfy clothes to wear, you really need to be on your toes with your spells and abilities to battle alone. To partially make up for your downtime, your travel time is greatly reduced by the ability to teleport to major cities any time you like (as soon as you're high enough in level, of course). In groups, Mages are in demand for their DPS and crowd control. Don't discount the utility spells as well. The buffs, food and drink and making convenient portals at the end are all highly valued by groupies.
Druid: I've been playing a Druid since open beta. I love the class and have Night Elf droods abandoned across multiple servers -- once I went Horde, I never rarely go back. Druids are great for soloing as either Balance or Feral spec and are usually welcome in any group while leveling. Because of their multiple forms, Druids rarely get boring and make traveling speedy even at low levels. Druids are also extremely versatile at max level. Groups will want you as a healer, tank or DPS. The only drawback to playing a Druid is that they are not easy to just pick up and play (unlike DKs and Pallies). You really need to learn the nuances of all of your options and it can be hard to go back to after a long absence from the game or playing another class. Many don't consider this a negative thing about the class, however, and enjoy the extra brainpower involved in playing.
Death Knight: Death Knights are cake to solo. Not only do they start at level 55, but they are practically indestructible and mow through mobs speedily. They pair up nicely with any other class for those lucky enough to have a quest buddy. DKs also rock the endgame. They make great tanks and are awesome melee DPS. Their main drawback is that they are a dime a dozen. Once you make it to max level, you'll find many others like you competing for instance and raid spots both in your guild and in PuGs. I highly recommend a DK to every casual player as an alt, but you may find getting groups too time consuming for your busy schedule.
Hunter: The Hunter is one of two classes that I have never played past level 11. It's just not my bag. They are, however, arguably the easiest class to solo. The fact that they have no non-profession-related healing abilities is countered by the fact that their pets take most of the damage and the mobs drop quickly due to massive DPS. Hunters are so easy to solo that they tend to be the class of choice for gold-farmers (do they exist anymore or do they just hack and steal now?) as well as children and players new to MMOs. Because their damage dealing skills are so powerful, they are definitely wanted in the endgame. But you have to have the spec, gear and skills to be able to function in groups and raids and 80 levels of soloing isn't going to get you these. Expect to have to work hard on your gear and grouping skills before being invited to the good PvE group activities.
Paladin: Before Blizzard "fixed" Retribution, leveling a Paladin was easy, but kinda boring and with a lot of downtime for mana regeneration. Now that Ret Pallies are OP (which I think is pretty much an uncontroversial statement), leveling is a breeze and a ton of fun. With barely any downtime and usually multiple mobs at once, I tear through content on my formerly boring, previously mana-challenged paladin. Also, when I have to make sudden AFKs due to my main job of taking care of The Spawn, I rarely come back to a dead Blood Elf. (Though I run the risk of being run off cliffs ever since I taught her how to run around in game. "Oops!", she says.) And once you get to the endgame, the world is your oyster. You'll be wanted as either a tank or a healer by everyone and be favored for (though not guaranteed) DPS spots as Retribution.
When making your decision as to what class to play, think about what role you want to fill when you are ready to participate in the harder group activities. Regardless of how you like to play, you are more likely to get into PuGs and scheduled raids if you are speced in a group-friendly way. So if you are unwilling to play certain roles, choose a class accordingly. Are you against both healing and tanking? Then pick a Hunter or other class that is completely unable to do either. Are you willing to tank, but find playing the whack-a-mole heal game a chore? Then choose a Warrior or Death Knight. But if you are happy to be flexible until you are able to get into a group that allows you to play your favorite spec, then a Paladin or Druid is for you. Taking either class to the endgame will allow you more opportunities and faster grouping than any of the other classes. And for the first few encounters, you will probably want to try healing. Groups are much more likely to be lenient on your mistakes (which you will definitely make) and poor gear if you are a healer, more than any other group role.
When it comes right down to it, no one can tell you what your favorite classes are to play. So if you pick a class that is tougher to level, but you enjoy it more then you have made the right decision. If you are torn like me, however, the above guidelines should help. What did I decide? Well, like I said, I'm a proud altaholic, so I picked two. My Pally is working her way through Outland and my Druid is thoroughly enjoying the quests in Northrend. My guild leader has promised me a healing spot in the second string 10 man raids once I hit 80. Of course, the fact that I'm married to him may have biased his decision a bit. But the healing gap helped too. Anyway, I'm off to play with a rejuvenated sense of purpose after my (kinda) decision. May you have as much fun as I'm having.