Some professions work quite nicely with some classes. You may find that complimenting your class with a profession makes leveling and even the end game a better experience.
For the sake of making Gold, its always a good idea to have one of your two main professions be a gathering profession (Skinning, Mining, Herbalism). Take two crafting professions if you want, but leveling them will require a lot of gold.
A word about secondary professions: level them up! Cooking, Fishing, and First Aid are vital to every class. Any well-rounded characters will have high skill in the secondary professions.
Druid
The leather-wearing Druid typically will benefit from most of the professions available to her. Tailoring and Blacksmithing are the exceptions, they are expensive, and the Druid can't wear plate or mail, and shouldn't wear cloth when leather has better armor bonuses.
A Leatherworking / Skinning combo works well, allowing the Druid to make much of his own gear, and do the skinning to provide materials for it. Skinning is also profitable, many of the leathers made can be sold on the Auction House.
Another effective combo is Alchemy and Herbalism. Alchemy potions can be used to make up for a lot of a Druid's weaknesses. A tanking bear Druid may not have as much health as a warrior, and a casting Druid won't have as much mana as a Priest. Potions can make up for those differences.
A Druid is a great harvester of herbs. With the ability to stealth and dash, Druids can get in areas too high for the current level and harvest herbs. And Herbalism is a great compliment to Alchemy.

Mage
Many of the professions work well for the Mage. Blacksmithing, Leatherworking, and Skinning will do you the least good, since you'll not be able to use what you make. Engineering and Mining don't have all that much offer when compared with some of the other professions available.
Alchemy/ Herbalism is great for a Mage, the potions offer great buffs and life-saving injections of mana and health.
Tailoring is great for the cloth-wearing Mage. You can make some of your own gear, not to mention inventory bags.
Enchanting is good for every class. The Mage benefits from armor and weapon enchants, and oils that increase spell damage.
Paladin
Paladins can wear Mail, and then Plate armor. That means that Blacksmithing/ Mining is a natural profession combo for them.
Enchanting is also a good choice, with picking a gathering profession (Mining, Skinning, Herbalism) as a second choice.
The other professions such as Jewelcrafting, Alchemy, and Engineering will benefit the Pally.
Tailoring and Leatherworking won't do a Paladin much good.
Priest
Priests are squishy clothies, so picking up Tailoring is a good idea. Being so mana-dependent and fragile, Alchemy is a great choice to make a Priest a little more hardy.
As with the other classes, it's a good idea to have a gathering profession, just to be sure you can always make gold.
Enchanting, Jewelcrafting, and Engineering are great choices for a Priest to have as a profession.
Blacksmithing and Leatherworking won't do you much good, unless this is an alt and you just want to try those professions.

Rogue
Rogues are leather wearing poison brewers. Herbalism, Alchemy, and Leatherworking are natural profession choices (and make sure you have your Cooking leveled up.) If you intend to make your own poisons, Herbalism is a must.
As a second choice, Blacksmithing to make incredible weapons is a viable option. Enchanting and Jewelcrafting make ok secondary choices.
Tailoring won't do anything for the Rogue.
Shaman
Shaman wear mail and use a variety of weapons, so Blacksmithing is a good choice. Other than that, the majority of the other professions will benefit a Shaman. The hybrid nature of a Shaman makes them fairly flexible in profession choices.
Tailoring should be avoided. Unless you just want to try it. You're paying for the game, so play it how you like!
Warlock
Gnome Warlocks (and all Gnomes) have a racial bonus to Engineering, so Engineering is a good choice for Warlocks. So is Alchemy as a way to self-buff.
Jewellcrafting, Enchanting, and Tailoring are good second choices.
Stay away from Leatherworking and Blacksmithing.
Warrior
A warrior ought to go with Blacksmithing/ Mining to always ensure he has the best gear. Alchemy/ Herbalism is great to make Rage potions.
Jewelcrafting and Engineering are always helpful.
Leatherworking, Skinning, and Tailoring won't do a Warrior much good.