GothikOrk: If any HoMM veterans can share their personal strategies and tips here to get me started that would be helpful.
Artki: Getting started can be confusing. Make random maps (small at first) vs one computer opponent and play just a week or two. Then quit, ,make another random map and play a week or two again. Do that many times.
You'll get used to exploring, how to use level 1,2 and 3 troops, gathering resources with one hero while another one runs around fighting monsters - the basics.
You should be able to play one of these micro-games in less than an hour and you'll be able to make lots and lots of mistakes without worrying about how they'll effect you chances of winning - because you aren't trying to win!
The only drawback to this is that you won't appreciate the impact of high level magic - but you'll learn that when playing campaigns and stuff. Just remember that expert level Earth magic allows you to cast expert Mass Slow - which is key for winning seeminly hopeless fights against giant armies of monsters.
And i think the most important ability is logistics! Being able to cover more ground makes everything better. The faster hero gets the artifacts first, fights more monsters per turn thereby getting more experience and leveling up faster. Gets back to town to get the latest army upgrades - the longer the game (or the bigger the campaign) the more important logistics is. Like high level magic, you won't notice it when you only play a week or two of a game.
While Logistics is not a BAD skill to have(unlike Eagle Eye which is utterly worthless and a waste of a skill slot), there are several that are far more important to have.
1)Earth Magic - One should get this one ASAP. The only way to cast Town Portal in any way to be at all useful is to have expert Earth Magic. Also spells like "Slow", "Resurrection", "Animate Dead", and "Implosion" are greatly improved with Earth magic skill. These are more or less the key spells in the game(the others being "Dimension Door"(AIR), "Fly"(AIR), "Haste"(Air), and a handful of the Water magic spells but none of those are as important as their Earth magic counterparts(re: Town Portal is much more valuable than any other spell in game).
2)Balistics - Without this skill your ground troops will be stuck outside of the castle walls when you are attacking a town/castle, waiting for turn after turn for all the walls to be destroyed and then finally the drawbridge to come down. Unless you are enormously powerful you will lose troops by the score due to enemy towers and ranged units.
3)Air Magic - Need this one to cast Dimension door multiple times in a single turn, as well as cast "Haste" on ALL your troops with a single casting, rather than one at a time. Also spells like Chain Lightning are much more effective.
4)Water Magic - Spells like "Bless", "Teleport"(I think this one is a Water spell?! Been a while since I played...), "Frost Ring"(which I rarely use but it can come in handy sometimes), "Prayer" and such are much better with Water skill.
There are a few more that are arguably better than Logistics but I will leave that there for now.
Now for SECONDARY heroes(re: not your main hero but the ones you buy/recruit afterward), you will want to have at least one or two "Scouts" whose job is to explore/reveal the map and collect whatever resources or capture any mines they can manage to get. Scouts should have "Scouting", "Logistics", "Diplomacy"(to recruit stacks of neutral armies who would otherwise want to fight or run away) and the "Visions" spell(to see which stacks are willing to join and which are not).
Other skills that go well with scouts are "Estates"(not much for a single hero but having four scouts with Estates will net you a few thousand extra each day) and(on 'Water' maps w/ Islands) "Navigation".
I am probably forgetting one or two skills here so bear with me.
Also it is a good idea to have one hero whose job is to be garrisoned at a(or at EACH) town you own. This garrisoned hero should have "Artillery" skill(gives control over the towers so you can aim at which of the attacking units you want to whittle down), "First Aid"(heals up damage to creatures and gives control of first aid tent, provided you have a first aid tent), Whatever magic skills you can get, as well as skills like "Armorer", "Sorcery" and "Archery".
Of course the garrison hero becomes unnecessary after your main gets the town portal spell with expert earth magic(he can just portal to every town he needs to to destroy would-be attackers).
I wrote a novel on Heroes III strategies and build orders and such a while ago but accidentally deleted the post before I was finished with it. Maybe if I find time later I will re-write it and post it here.
A few quick notes:
"Conflux" is a sort of 'Cheaters' town". One almost cannot be beat when starting with this one because you can be cranking out 4 7th level units per week starting at the beginning of week two if you really want(all other towns limit you to 2 or 3 7th level units per week and you won't be building them until a few weeks in at least).
"Necropolis", and the Necromantic heroes and units are fantastic! The one faction that can have legions of troops within a week or two if things go decently. Give a necromancer/Death Knight the CLoak of the Undead King as well as the armor of the damned and the game is over.
Do not bother with Ammo carts. They only provide targets for Magogs and Power Liches to wipe out your troops.
EDIT: And yeah, what Mystal said above about map sizes and specifics is true. "Logistics" is never a good choice on a small map or one that is mostly water for example.