At the dawn of World of Warcraft Engineering used to be a profession mainly focused on its usefulness specifically for PvP, (Player vs Player). But that is not the case anymore. This profession has had a lot of changes in the past few major content releases of World of Warcraft with a lot of cool new additions to attract all kinds of players in general. Epic head gear, useful trinkets, guns, bullets, arrows, scopes, ability to use cool gadgets, portable mailbox/repair bots are some of the things that come to my mind. And not to forget, "THE EPIC Mechano Hog"

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Engineering is best coupled with mining as your secondary profession similar to
Jewelcrafting which also benefits a lot from mining, unlike
Inscription or
Alchemy which benefit a lot more with having herbalism as your secondary profession. The reason is quite obvious here. You would need a lot of ores/bars of different kinds varying from old school stuff in
Azeroth upto
Northrend. The sheer amount of gold required to be invested would itself make it a smart choice to go with mining as your 2nd profession, although it's not mandatory if you have the money supply coming from one of your other characters to fund you through.
Taking up Engineering as your profession not only allows you to make cool gadgets, bombs etc.. but also craft your own mounts to travel around the world, be it ground mounts or flying. Also similar to some other crafting professions like Tailoring or Leatherworking, Engineering also gives you an option to choose a specialization which are either Goblin or Gnomish. Each of these has its own advantages and disadvantages which we will discuss in detail later. But it's not going to be a game breaker if you choose one over the other.
Having said that much for introduction, let's get straight down to business...