First step: remove the old fixture. While removing the glass cover, Lisa discovered that it had multiple cracks in it and was surprised it hadn't fallen out of the bracket. She put the broken cover in the trash.
Until Lisa removed the broken-glass cover, we'd never even known that there were two interior light sockets here, as we'd only changed the outer ones by reaching inside the fixture. But as we saw when Lisa took the cover off, removing it possibly would have had the cover go pieces in our hands anyway. Lisa saved this fixture (without the broken glass, of course) for possible re-use elsewhere.
Lisa shut off the circuit breaker feeding the master bathroom and removed the old fixture, revealing the earlier color of the bathroom wall. Repainting the bathroom (probably a similar white, but the paint could stand to be renewed no matter what) is on the list of lower-priority tasks for this room.
With a tiny bit of help from me (I hold tools and parts sometimes), Lisa installed the new long-tube florescent fixture, which uses two 24-inch T8 (the smaller type) bulbs.
With the protective wrap removed, the bulbs installed, and the diffuser cover in place, it was time to re-energize the circuit and flip the switch. Success the first time!
The new fixture produces 2650 lumens for 26 W, compared to the 700 lumens for 34 W we were getting from the old fixture. That's roughly 3¾ times the illumination for ⅔ the energy, which seems like a really good trade in my opinion. And I reckon eventually the electricity and long-life-of-bulb savings will pay for the cost of the new fixture, but that wasn't the main point of the purchase. It was just that we were tired of bumbling around in the dim.
It's a big difference in the bathroom, and I'm glad Lisa had us buy the "warmest" tubes, as the "cool" ones would have probably burned out our retinas in such a small space.