Top 3 and most costly mistakes I see in most MP games:
1) Not scouting. This one's huge. You need to know the terrain before you start popping settlers, otherwise you'll often end up blocking yourself from the best spots because you settled in what appeared to be a good spot, but is in fact 3 tiles away from where the real money spot is. You also will fall behind in tech for not meeting other players. You will see this a lot with builders who get discouraged and quit in the renaissance era as everyone else's score suddenly skyrockets. That's because they met the other civs and are benefitting from a huge tech boost for knowing more civs. Exploration is a core pillar of the game, so even if you insist on building a monument first (it's fine, I do that too on archipelago) you need to build a scout 2nd. Nevermind all the goody huts you'll miss out on, and the lost freebies from city states etc etc.
2) Building too many wonders. Well you can see most people agree this is a fail tactic. The player who always has a wonder queued in his capital NEVER, EVER wins MP. Pop a lot of wonders you are doing nothing but hindering your ability to defend yourself and/or settle new cities. And you are making yourself a target. What's better than spending 30 turns building a wonder? Spending 10 turns building troops and getting the same wonder. And then still having the troops.
3) Inadequate defenses/not planning defense into city settling. This one is also way too common. What is the point of building all those sick cities if you aren't going to defend them? What is the point of putting a city on the coast where it gives so many tiles of water access, it can be oneshotted by a few frigates and a couple privateers? If you absolutely must put a city with 5+ water tiles beside it to get a resource or something, at least throw a few archers around it and stick a couple triremes out front. Yes, a lowly trireme in the right tile will save your city from one turn sieges once the navigation ships come out. The other thing is, put troops outside your own borders to get advance notice of an incoming invasion. It's far too easy for me to slowly assemble a massive army 3 tiles away from somebody's capital, when they don't put some pickets around the borders. It usually takes 10, 12 turns to move all the units over there, but if I can just put them to sleep literally 3 tiles from your capital until the rest arrive... you are in trouble. All you had to do was put an archer on a hill around the edge of your border, or park a trireme on that workboat, and you woulda seen it coming. I'd be forced to attack with a weaker force or form up further away, also buying you time and giving you more notice to defend. Sadly, it's far too easy to just build up 20 or 30 units beside someone's city cause they literally never look outside their own city for some reason. Just gift your cities away if you're going to play like that, cause there is no point founding cities and just leavin them sitting there like ducks.