How to Introduce project management to an organization?

Julie,

i do also work in a PMO which was build up from scratch. While all what my fellows written here is absolutly true i could just strongly recommend to follow Mounirs advise.
Set it up as a project itself and get professional and experienced help from outside since i understgand that you and also your supervisor have no experience.
Yes, that will need an not so small investment by your company but if you could answer all the question my fellows written here it should be easy to convince your senior management.

Best of luck!

I hope this information helps:
I work in a department titled, Department of Philanthropy, Engagement and Project Management. Below are excerpts from our department’s operational document.

From the Mission statement regarding PM:
• Providing leadership, direction, support, accountability, centralized coordination and execution of comprehensive cross-divisional projects and collaborative efforts across the Noble Foundation,
• Assisting in the optimum leverage of available resources, and
• Assisting in the alignment and prioritization of projects and programs.

Part of the Value statement:
The value of PEPM is that it provides the Leadership Team with a core set of practices to consistently evaluate, prioritize and execute projects and collaborative efforts that will have the greatest impact and align with the strategic objectives of the organization.

Primary responsibilities for PM:
Project Management
• Build a core set of practices that deliver consistent processes, oversight activities and tools used to plan, create, evaluate and timely communicate comprehensive, cross-divisional projects including, but not limited to:
o Facilitating a purpose driven approach utilizing a scoring criteria to ensure that projects align with the strategic plan and priorities approved by the Board of Trustees
o Designing methods to track project status, milestones, deliverables, issues, risks, etc. and create individual project reports as well as a consolidated overview of all projects.
o Collaborating, coaching and motivating project leaders, coordinators and team members on a proactive basis in an environment of transparency and mutual trust,
o Working with staff Economists to create processes for estimating project impact and benefit our consultation area.
o Determining how results will be measured and completing post-project evaluations to determine how well results were achieved
• Gain an understanding of Foundation-wide resource capacity and inform the Leadership Team of resource constraints that could possibly prevent the successful implementation and/or completion of projects,
• Actively monitor projects, with regular reporting to the Leadership Team to allow them to make decisions regarding consolidation or elimination of projects not providing strategic benefits,
• Facilitate selection, implementation and use of a project management software package,
• Provide administrative support for project delivery, and
• Obtain and act upon project lead, coordinator and team member feedback.

Below is what is described on our organizations intranet page:
The Project Management Office (PMO) was established to focus on developing, implementing, and supporting the use of proven project management methodologies, tools and techniques in order to meet or exceed the needs and expectations of a project and the stakeholders while ensuring projects are aligned with the organization's mission and strategic goals. The PMO also assists with transparency and visibility of project status to team members and executive management through reporting and appropriate communication methods. Other PMO duties include capturing project requests, facilitating the project selection and prioritization process and monitoring ongoing project and program performance.