Curently, I'm a sophomore taking up Business Administration - Management but I used to take up Accountancy although I'm not very much of a fan in finance and I'm not very much of an interactive person so I think Management doesn't really suit my personality. I'm thinking about pursuing my dream as an engineer. What could be the best Engineering path I should take?

Janie, think about the industry that really excites you every time you name it and pursue the engineering stream that is relevant to your inner passion. You can choose from many different engineering streams such as mechanical, civil, electronics, computers and so on. The broadest of all streams is mechanical engineering, as the knowledge can be applied in a variety of fields from automobiles to smartphones (design to manufacturing) and keeps your options open in future. Of course, the other streams will present unique opportunities that you may not be able to pursue with mechanical engineering knowledge.

Introvert personalities (including me) can excel in any given field and some famous personnel are Albert Einstein (science), Abraham Lincoln (politics), Bill Gates (software/hardware) and so on. Check this link: http://www.huffingtonpost.com/2015/08/15/famous-introverts_n_3733400.html.
Management is not only for extrovert personalities and don't be hard on yourself about the personality match with the management degree.

The last part, everyone including me want to be rich and support the family. So do not worry about it at this stage and prepare yourself to choose a career path that matches your interests (work, life balance, job satisfaction and monetary terms). If you need any information, do not hesitate to message me.

The key advice I have is, if you want to be an engineer, get into an engineering program. The BA-Mgmt track will not make you fit to be an engineer at all, so there's no point "wasting" time with that during your undergrad program. I'm assuming you are a sophomore in college already unless it's a "BA-Management" class that's offered in highschool. If that's the case, I wish I had gone to your highschool.

Regarding your desires for the future household, a lot of people can successfully run their households and plan for their family's financial future without having to get a degree in finance, accounting, or another related field. Financial success is as much a result of personal choices as it is the outcome of your education. A lot of people with only their GEDs can save up enough for early retirement; and a lot of highly educated people who work in finance can make very poor decisions.

Some good news is that, as someone who has spent 7 years in engineering design (it paid the bills even though I wasn't looking to become an engineer) and is currently in finance, there are a lot similarities between the two fields in that: 1) both tend to be very detail oriented with a focus on long-term results, 2) both are very numbers-based, 3) if you have the mentality of being a problem-solver, and enjoy doing that kind of work, the only true limiting factor you'll ever have, as far as "success" is concerned, will be yourself.

As Ken suggests above, the first step is to see if engineering is a good fit in the first place. I'll withold my lovingly-snarky biases about engineers, but one thing engineers can do pretty well is recognize the goal or problem, then figure out the best resolution to reach the goal or solve the problem. If you're dream is to become an engineer, regardless of what classical discipline it may be, focusing on accounting, finance, or management will probably not serve you too well at the start of your academic or professional career. That being said, though, there are a LOT of people who get an undergrad degree in engineering, then go back to school for their MBA or a graduate degree in a business, finance, accounting-related field.

If you want to be an engineer, go to engineering school and become an engineer intern then a professional engineers license. If you want to be an accountant get a degree in accounting and become a certified look get a degree in accounting and become a certified public accountant. The CPA may require a Master's Degree.