In order to answer this - I'm going to address firstly your 3 points and then some broader advise. For the record, I work in IT.
"Making people support me" - This is probably the biggest one - the question that is being asked here is one of leadership. I don't mean the barking orders at people and bossing them around, I mean personal leadership. If you set standards for yourself and adhere to them, if you demonstrate to others that you are dependable and will follow through - then people will naturally gravitate towards you. And don't think that this means you have to be perfect all the time - One time, I was working on a project and I had a junior do some analysis work, I looked through it - it looked good and I signed it off. We went to do the change and we had missed something big, which caused a bit of noise. I sent an email around the office personally apologizing for the issue and I bought a cake. The ownership I took and not throwing the junior under the bus was very well received - because people saw that if they worked with me, I would treat them fairly.
Depending on your outlook on life - I can highly recommend the Extreme Ownership series by Jocko Willink (ex Navy Seal and ultimate badass) for more on this subject.
In short - lead yourself and then others will want to be lead by you.
"Exerting influence" - A big part of this is what I've addressed above - however in addition, you need to think of selfish altruism - "I want X from someone, therefore I need to figure out what they want from me" - For me (in IT) helping out the finance team on various tasks and doing some automation for them, meant that when it came time for them to sign off my on-call, I didn't get any questions asked (yes, it was all legit - but we had a project that caused some high hours that normally would be questioned). If you make others lives easier, they will be more inclined to help in return.
"Being knowledgeable" - The short answer here is that there is no shortcut. In order to be seen as knowledgeable, you gotta be knowledgeable. In addition though, this means (as you said you are introverted) you have to get more comfortable with speaking up in meetings and making suggestions. Don't worry about making a wrong suggestion either - an incorrect idea that has sound supporting evidence is still worthy of consideration. Saying something like "I see the solution that we are trying to provide, but this looks similar to this problem we solved this way previously - is there a way that we can use the same principle to resolve this issue?" - it may not be correct, but if it's got a good grounding, then it shows to others that you are thinking of alternatives and offering potential solutions.
Furthermore, if you think that an idea has legs, you have got to be willing to fight for it. If you make a suggestion that gets rejected and you slink away into the corner, then no one is going to take you seriously - whereas if you back yourself and give your reasoning for this idea and why you believe it will work, people will take you more seriously. There is a balancing act here - fight too zealously and people will dismiss you, fight not enough and they will dismiss you - you need to remain calm and neutral and not invested in one way or another - but seeking the best option.
Finally - you talk about bullying and advancement - my 2 best bits of advice:
Learn to say "No". No is the single most powerful word, it lets others know that there is a boundary that you are not willing to cross. Use it sparingly and with caution, but when the time is right - use it. You may face a little bit of short-term pushback - but in the long run, it will work out for you - and if not, that's when the organization is so toxic that it would be better to look for other opportunities.
Email is your greatest friend, in the world, bar none. Seriously - get used to writing emails and following up informal chats with emails. Get used to searching your mailbox for key info. Say you talk with your boss to take time off at short-notice and they say "Yep yep, fine whatever" - and then deny the conversation ever took place - a follow-up email of simply "Hey, just confirming the chat we had earlier - as discussed I will be off work from the day after tomorrow because of a family emergency".
Hope that helps.