Technically, as long as you have a method to provide materials for whatever crafting professions you choose, you can viably take any combination of the various trade skills.
Even when someone takes blacksmithing and mining, they are short of materials for numerous of their recipes, for instance arcanite bars which only an alchemist can make.
If you have a large ammount of disposable wealth, then you do not need to take any gathering profesions to provide materials for your crafting profesions, you can simply buy them off the AH. This however is inadvisable for those without sufficient means.
Something that I did was level mining and herbalism simultaneously (a daunting task, but made slightly more mannageable with the gatherer add on) and saving the mats up on my many different alts until I had collected enough to power level engineering and alchemy simultaneously. I soon realized that I was less than impressed with engineering and dropped it to re-level herbalism to provide some hard to get mats for a few of the pots that I wanted to get in the market in.
If you don't have the cash to level your crafting skills to the point at which they would start to provide you with an income, it would be best to switch to gathering professions until you either have the cash, or the mats.
Enchanting is a unique beast all together. It is fed by valuable magic items found in the game. You can either farm them in instances, craft them with your second skill, or buy them in the AH (which eventually gets really expensive). The second choice may help you find a possible second choice for another prof, however you don't want a second prof that requires mats that you can't farm yourself. This is why I recomend tailoring as a second skill since you can create items with the cloth that you can find from humanoids then turn around and disenchant those items to provide mats for your enchanting.
The enchanting/tailoring combo has proven itself while trying to level up my paladins trade skills. This method may still prove expensive because many of the tailoring recipes still require items other than clothes.