Don't worry too much about being beat to the classical age, happens to me almost every single game. If they are the first to industrial that's another story
Agree with others saying archipelago maps are not the best for training. They are too easy for the most part (the AI sucks at naval invasions still). Try continents for the most rounded experience.
Prince is not a terribly hard level, just try it out a few times. If one thing doesn't work, try something else. At that level you can pick any number of potential strategies and be successful, you don't need to play "ideally" to win.
General tips for your level:
- grow your capital as large as possible in almost every circumstance and build all the national wonders possible there
- trade extra resources to the AI for lump sum gold (300 every 30 turns if friendly)
- try to stay ahead of hapiness by 5-10 points or so to keep growing
- don't forget city states. When policies are coming in too slowly, try to grab a cultural CS ally. If you have a large happiness buffer, grab some Maritimes to grow faster instead. If you find yourself having more than a couple allies, use the Patronage tree (first 3 policies on the left side are awesome)
- Plan your general strategy and keep in mind the policies that will be available at that time. For instance, a common strategy is the REX (wide) approach. This will found many cities and eventually happiness will be a problem, so after Liberty the best option might be Piety, in order to get Theocracy. Then you will have a large empire and Order will make a lot of sense when you hit industrial. Planning out "Liberty-->Piety-->Order" at least gives you a framework for your general strategy to play off of. Don't be afraid to adapt if needed, but in general you don't want to be thinking too hard about a policy choice the turn you get it - that is a sign of poor planning.
- learn some of the more common strategies of the early game. It's important to get a good start to set yourself up for the rest of the game. There are tons of articles on here about the NC Start, legalism/meritocracy usage, Tall vs. Wide starts, etc. Learn the steel slingshot/longsword rush - it's ideal for taking out your nearest neighbor. Everything is situational, but you need to have the tools to react to whatever happens.
Keep reading the boards, there's tons of good stuff on here for all level of players. Good luck and have fun.