There are a few things you can do to accelerate levelling:
Dungeons
Use the Looking for Group tool to queue for dungeons. There are often quests within dungeons with good experience rewards (the first time at least). Dungeons also have high level mobs which will have a good rate of experience on them. Loot from dungeons will also make your character more powerful, making questing easier/faster.
Quest wisely
Move out of areas once you out level them - don't feel compelled to stay and finish the story lines if your priority is levelling.
PVP
Some people get a good rate of experience/hour by winning in battlegrounds.
Have a max-level character Your colleagues will have advantages over you if they have access to Heirlooms. As well has having excellent stats some heirlooms have bonus experience for both killing mobs and turning in quests.
Rest bonus Make sure you are in an Inn or a capital city when you log out to accrue rest bonus.
Gathering professions Nowadays you gain experience when you gather herbs and mine ore. You can then sell the herbs/ore on the Auction House and buy better items to level faster.
Guild Perks Probably the easiest action for a new player is to join a guild and take advantage of the levelling perks, especially Fast Track (lvl 6 guild). Other useful perks for levelling are Mount Up (lvl 3), hasty hearth (lvl 8) and working overtime (lvl 14). wth thanks to @Jereone for the the reminder
Best Answer
Regardless of which guide you follow, the cheapest way (when only looking at gold as a cost factor) is always going to be to take up Herbalism as well or have an alt with Herbalism.
Using your favourite Inscription levelling guide and Wowhead for example you can find the best routes for specific herbs and farm away.
Some addons such as Routes and Gatherer also help improve the speed in which you farm your herbs.
That doesn't mean it is the fastest but that would be the cheapest way as it costs you no gold.
However, there is an argument whereby time also is a sort of cost. Due to that, a lot of people find it more cost effective to do other tasks which take less time that make the gold you need to buy the required herbs for levelling Inscription with spare change and time left at the end.
But that is another whole topic in it's own I think.