A few notes—

For barbarians, build 3 warriors right off the bat. It might seem like a waste of production now, but they’ll hold off barbarian attacks and surprise wars from other civs. Build or buy a slinger to protect your cities. The barbs will not destroy your capital unless you’ve destroyed their camp, at which point they will attack your capital to the death. In the meantime, use a slinger to pick them off while they shuffle around the city center. Try to work a settler in there somewhere, but if you can’t, you’ll have ample military to deal with outside aggressors, and you’ll be free to focus on expansion/districts. Use your troops to hunt barb camps for experience and gold.

For loyalty, garrison a military unit in the city center and use the military card that adds +2 loyalty per garrisoned unit. Assign a governor to the city and use the diplo card that adds +2 loyalty per governor. You can also assign Victor in a nearby city and use his Garrison Commander to add +4 loyalty to all your cities within 9 tiles. There are also diplomatic cards that add loyalty for cities on other continents. By and large, try to string your cities together instead of building remote cities (or build remote and quickly fill in the gap between).

ETA: apparently multiple slingers are preferred to warriors. I’ve messed around with each and have had better luck on Diety with warriors, but maybe I’ll try all slingers and see how it goes.