How to best use high ranking connection

I am a recent graduate and found of a family friend with a personal connection to someone in the quite high ranks at a very big financial institution (think MD, central Europe).

While that's a quite hopeful coincidence, as I could most likely get a good introduction and get them on a call, I am unsure as to how best proceed once that happens and what I can generally expect from them. Obviously I want to ask for their help in eventually getting a position at the company, and need to make it obvious but I also need to find the best way to "help them help me".

Still, would just an inside reference from them or someone they connect me with significantly boost my chances if I still go through the Graduate channels? Could I possibly ask to skip these channels completely? (note that I have 3 years of exp at B4 audit/reporting already).

Any insights or experiences would be very appreciated.