I started with Lightroom Mobile on my phone. While it doesn't have all the features of the desktop version, it is capable of producing perfectly serviceable pictures for sharing on social media. I'll typically use this for updating friends and family from the road and then do more comprehensive development after I get home. It even syncs the adjustments between the two - mobile and destkop!
I use the Samsung Galaxy S8 with a 64GB internal microSD card and a USB-C SD card reader. At first, I simply plugged in the camera's SD card and imported the whole lot into Lightroom. It worked for the first trip but it's not quite the solution I wanted. So I bought a USB-C hub and portable external hard drive. Now I can connect the phone, camera card, and hard drive onto the same bus. I could even connect two hard drives to save duplicate backups of all my pictures. I took this kit on the road to try it out.
It didn't go well.
For a techie guy I can be surprisingly impatient sometimes. With everything connected, the phone simply wasn't recognizing the SD card and HD as external devices. I don't know why. After several attempts I gave up (for now) and connected the SD card directly to the phone and started a file copy. It was a slow copy, a lot slower than I expected, so I put it aside and came back after a nice brunch and walk around town.
It failed half-way through.
I forget the error, I was tired and didn't take a screen shot. I tried again and got the same error again. I had a little over half the RAW files on my phone now and no time left for a third attempt. We were back on the road.
Then it happend. Google photos decided it ought to upload these photos to my account. These are RAW files, most of them garbage. By the time I realized what was happening my battery was dead and AT&T was sending me annoyed alerts (yes we know you have an unlimited data plan, don't make us slow your upload speeds). My photos account now had a lot of RAW files I had no intention of keeping, when I got home I would have to go through and delete them all. This kept getting worse and worse.
I still think the idea is sound. At the end of each day I could, theoretically, save my photos to two hard drives (as a precaution against theft, for example, carry one with me, leave the other at the hotel), quickly develop a few gems as I go, and have everything ready for more intensive processing when I get home. It's a great idea. I just need to dig in and make it work.
In this case, I shared a few photos directly from the camera, which also works. At least my Facebook friends thought it was okay.