I've gotten confused on this problem way too many times.. I'd like to
get some definitive starting points.

When you see net adapters online they are always rated like

10/100  or 10/100/1000.  So how does one turn that notation into
megabytes?

That notation is in megabits per second, or Mb/s or Mbps. Bits are shown as lowercase b and bytes are shown as uppercase B. So you want to change Mbps to MBps. This is stating the raw throughput, so Ethernet headers are included.

 
But still, when I'm trying to measure how much data is moving to a
certain directory, and I want to compare it to what the adapter is
supposed to do... (in some easy homeboy way).

I vaguely remember something about 8 bits to a byte or maybe its the
other way round...

Yep - 8 bits to a byte in this case. Serial comms can be a different number of bits per byte.

My homeboy transfer measurements:

I measure the incoming MegaBytes as measured with `du' with a while
loop interating in settable intervals.  So in this case when set to 60
seconds,I now the number of megabytes that arrive in 60 seconds but
would like to know how to convert that to the other notation.

du is probably not a good way of doing it, depending on how the disk system commits the writes. Some clients show you the data rate. Maybe wget or ssh? Cant check where i am now. Thee numbers the application states (or du) is just the data, so doesnt take into account ethernet, IP and TCP headers.
 

I'm seeing between 222 and 237 MB in a full minute being transferred
and it seems quite slow for what is supposed to be a gigabyte network.

This is just across two computers on my home lan, both with gigabyte
adapters and they connect thru a gigabyte switch.  Or I hope they are.

My setup looks like this in brief (simplified).



IIRC typical speeds on 100Mbps LANs are 4 or 5 MBps. There's many factors that can affect speed tho.