My thought was a thread dedicated to tips for players who aren't brand new would help a lot. If you've got the basics of combat and building down then these tips should help.
First, to start off, some players are reporting that setting the AI to Multiplayer is making the AI play at absolutely max difficulty and it loves to rush, so I'd suggest not using that if you're new.
If you're newer to the game I'd suggest trying out maps which are safer, to be able to get more used to the economics of the game. Helltide has been my personal favorite, it's a map where the players cannot fight until the final battle is called and the game ending fight happens, anywhere from the end of Week 4 to Week 5.
Easy Towns: Temple, Dungeon
Intermediate Towns: Schism, Necropolis
Hard Towns: Hive, Grove
As any Town, it's pretty much a must have to rush down the laws that give you the one time Wood/Ore/Gold especially early game. Additionally, any laws that give you additional law points, resources per turn or gold per turn are very good to take immediately.
When selecting a Hero for your town, it's a good idea to have a loose plan in mind. For instance, the Mercury Hero for Necropolis is fantastic for rushing Vampires, but for rushing Dread Knights not so much.
The Hero Sub-Classes that are available are mostly a consolation prize for making a bad build as it currently stands. I highly suggest going for skills that help you now or help you soon, rather than shooting for one of those more unique bonuses. YMMV depending on starting hero, Schism's Icequeen for instance with the minimum damage one is easy to reach for instance.
It's important on every hero that's going to be used for fighting to either have 2 Schools of Magic or taking 1 School and Wisdom. All Casters should take Thaumaturgy, all Might heroes should strongly consider tactics.
Something that made a massive difference in my matches was realizing how consistently many of the towns can rush out a Tier 6 or Tier 7 unit by the end of Week 1. It's very important to go for more units early. Going for a full economics build in Olden Era is VERY risky in comparison to how HOMM3 was vs AI.
Familiarize yourself with the spells in the game, particularly Tier 1 spells. Mouse over the spells in the Guild or your spellbook and hold shift to see what the upgrades do. For instance, Shadecloak is pretty great without any skill into it, but spells like Primal Hide, Unnatural Calm or Blessing get really crazy once fully upgraded. HOMM3 is dominated by Slow/Haste spam, but there's a lot of better low tier spells in Olden Era.
Don't forget to get some Astrology Points in your Town Hall upgrades. Not having access to 2 charges of Town Portal later on is really crippling in a pinch.
Due to Laws being able to give injections of Ore/Wood, only finding one of your Sawmills/Ore Pits in the first few days isn't so bad, don't waste multiple turns searching for it when you have better things to pick up on the ground.
I've noticed a lot of posts where players are struggling against early strong AI armies. Feel free to post images of those too and I'd be happy to describe what to do to beat it.