what do you folks generally do to optimize beddy-bye time?

Most of the time, I don't look at any screen.

Instead, I read (paper book, a journal or a magazine), chat with my spouse, sketch and journal. Earlier in the evening, we may decide to watch a DVD together, but that's not everyday. No coffee and seldom any tea late in the evening (or something herbal, then).

And almost without exception, the moment I put my head on the pillow I fall asleep. There have been many exceptions to that rule, mind you. That was back in those days where I was trying to escape my own personal daemons instead of facing them (like not getting rid of my own addictions and bad habits).

Like already suggested in the comments, if you have hard time sleeping melatonin is an option but on the short-term only. Don't make it a long-term habit. I also consider the extra sleep it may give you not the best sleep, no surprise you may feel groggy.

To the insomniacs, what are some things you do in the wee hours/early morning for a relaxing start to your day?

Since I don't need much sleep and I do my best work early in the morning, most of the days I wake up very early, around 3-4 AM and... I don't look at any screen either. Imho, it's even truer to realize that screens and online content are pure poison to the mind and to the soul early in the morning, like they're in the evening.

I drink a large glass of water and appreciate the quietness for a few minutes. I may flip some pages of a book but most often I just sit or stand still breathing slowly. It probably is the closest thing to meditation, for me.

After I shower and dress, I'll sit at my desk and start writing longhand—no screen remember, and no music either. Nothing but the still sleeping city around me with its otherworldly quietness, and me slowly scratching some paper with my fountain pen. (In summer, I'll open the window next to my desk or even go sit on our balcony so I can better appreciate that magical quietness, and a little later start enjoying the company of the earlier-riser of the few remaining birds, the ones that start singing their songs well before car/bus/motorcycle engines overrun their voice for the rest of the day.)

I will write up until we're getting close to the time my spouse usually wakes up @ 7-8AM (btw, we both work from home) at which time I'll prepare breakfast and, while she is getting ready, I often will go grab some fresh pastry for her at the local bakery (they're handmade and she loves them). We have a quiet but also chatty breakfast together. When she's ready to start working I will go out for my first walk of the day, in order to get my body moving.

Now, the normal day is allowed to start, with its constant noise and its unexpected and very often less than pleasant events. I'm ready ;)