I also find Monarch easy. The big jump for me is when the AI first gets an extra settler. It then expands so much faster.
Without the demigod levle, Emperor to Deity was the biggest jump for me. Dont know now...
In my current deity game, I managed to expand as fast as the AIs (they got to 3-4 cities when I had 1, but I got to 7 cities at the same time as them, and same with 10, 12, 15 cities. I had a 4 turn settler pump to do it.
The most important skill is knowing how to broker tech with the AIs. This is more important than anything else, imo. (Ok, if you dont know how to build a settler, that is more important).

Also, important in this is going and finding all the AIs. This often involves getting a couple early ships scouting in different directions, possibly suiciding through an ocean to find land.
In my current game, all 4 AIs on my continent found each other (with their tons of units) before I could find them. So they swapped all their tech first. I scouted to the other continent (70% water continents map) with a couple suicide curraghs, and found the other AIs. Those AIs on the other continent had Philosophy and Polytheism, which no AI on my continent had. I bought Polytheism for several hundred gold, traded it to every AI on my continent for a bunch of tech. Then, no AI on the other continent had MapMaking. I spend several hundred on that, and traded it to the AIs on the other continent for Philosophy and some other techs I was missing. Then I came back to the AIs on my Island and traded Philosophy for the rest of the tech and to recoup some money. I spend a few hundred gold to go from having gained only writing (10% sicence for 50 turns), to tech parity (near done with the age).
I then went and researched Literature. I held it to myself for awhile, and then swapped it for Currency, Construction and (with some money) Republic. Then swapped those to various AIs to get money, and was in the next age. As a scientific Civ (Sumerians) I now had a 1 tech lead, and due to my 4 turn settler pump, had parity in #cities as well.