Help! No development infrastructure in company! I've been tasked to implement it!
Have you been in this scenario before? 


Yes. Pretty standard scenario in my experience.


  How did you address and overcome these obstacles? 


In very small steps.


  How long did it take to implement? (as a rough idea)


Very long; in one rather large extreme case: over 10 years, and still ongoing.


  How would you prioritize the tasks listed above?


My priority, with additional comments:


Throw all active projects into source control.


This should really be a no-brainer and low-hanging fruit.

Implement coding standards.


Sensibly.

Introduce a methodology more suited to how the team currently operates, most likely Agile or Scrum and the tools and documentation templates that go with it.


If there is no methodology at all right now, then I'd start with a very lenient Kanban first - which basically just means throw up a ticket system and keep track of everything relentlessly. I would hold off with full-fledged Scrum until everything is on track and you're established as a Good Guy.

Implement staging so that we have Dev, QA, UAT and Live.


With this goes more separation; i.e. devs need to be weaned off of the production systems. 

Introduce tooling and setup training for development team.


Try to have the teams develop their own tools if at all possible.

Implement CI/CD
Bring all projects up to using the same technologies.


Might just not be feasible, if by technologies you mean languages and libraries. If you mean Docker or OpenShift or whatever, then sure.



The most important tip for you from my experience: avoid, at all cost, to do everything yourself. Give "impulses" to the teams, but they have to do it (whatever "it" is) themselves. If you do all the work (either personally, or with your separate "DevOps team"), then you will, at the end of the day, not gain that much as everybody will just depend on you constantly.