I like playing on prince myself it allows you the freedom to do what you want and with "better AI" checked on BTS it's actually pretty decent.
I've played a few monarch games and where as I could see myself winning it just takes to much micromanagment and planning it takes out the fun and turns it into a numbers game, I usually don't finish them after about midway, I just get lazy and want to automate.
Aslong as you understand the basics and the strats you can automate everything on prince and still have a tech lead, be in the top tier for score, and keep up militarily (with better AI).
The trick is just to understand the basic strategies, I often will use a financial civ and automate my workers because in BTS they will actually cottage spam after building farms for growth aslong as you dont have the "Dont change old improvements" checked.
It's getting kinda boring though knowing your gonna win.
Axe Rush pretty much equals Game Over.
Also, I think the biggest problem I had when I first started going up in diff was over-expansion or under expansion, not understanding how to rush very early, not knowing how to properly use slavery, not understanding city placement or specialization, you have to strike a nice balance between expansion, teching, and military (especially with better AI).
I'd actually suggest before moving up a difficulty switch your current one to Aggressive AI and see if you can win, if you can your usually good to go for a regular game on the next level, it really improves your gameplay learning how to adapt to the better AI, you just gotta understand that if you show a weakness they "will" go for it if they are an aggressive civ.
I always go for a Military victory and if your having trouble moving up I'd suggest trying Shaka out and reading up on early axe rush strategies but mix in a double stack of Impi to cut off his resources early and deny him copper, you use the UB to drop your city costs by 20% which can be huge and he has beautiful synergy with an early rush due to being aggressive and expansive, faster barracks, faster workers, faster granaries, Impi for double movement and cutting off resources, it really works well to get you a great empire, the double capital just seals the deal.
Try to learn how to do a rough dot map in your head just by looking at the terrain and counting the surplus and negative food, it's the single biggest thing you can learn to do, and dot mapping everything just takes to long, find a spot that would make a good prod/com/gp city and just do a quick count up of the +/- in your head to see what makes the best placement in the area.
Also, one or two extra gold or a gem/gold mine even a seafood+fin or an oasis in the early game can be game breaking, if you wanna rush moving your settler to a plains hill right next to your start location that can be huge even if you lose a turn.