Where to get started with monitoring?

When I started we had *no* monitoring at all, over the last year I've been able to find time here and there to monitor things of particular interest (NAS disk usage, server room temperature, basically things that have been a problem in the past...)
The monitoring is done on an ad-hoc basis with off bits of Python and Shell being piped through Fluentd and into Grafana. So you can see why I was so interested last week.

Onto my question proper: it is in my objectives this year to setup 'network monitoring' (yes, that ambiguous...) as we have experienced a number of issues. Having come from an enterprise background, monitoring and networking were done by separate teams so it's not something I've had any real/in-depth experience with.
An additional problem is that almost all of the network infrastructure is provided and managed by our company IT department, they provide us several VLANs to do R&D on. It is the IT department that manage the switches, gateways, firewalls, etc.

Could you suggest where I might start? We have a lot of smartphones, tablets, and laptops connecting to some NASs and various streaming servers. I think we want to monitor network quality (congestion, dropped packets, etc), how much traffic, and possibly store packets for a short time for analysis.
My understanding at the moment is that I need access to the switches/gateway to get this kind of information (which I am unlikely to get according to preliminary talked with the IT department), but are there other ways that won't impede traffic (i.e. I don't think I would get away with using a proxy)

Many thanks for the awesome show,
Ben