I finished M&M2 awhile ago, and had a pretty good time doing it. I just started M&M 3. My first reaction... holy crap all these icons everywhere are giving me a seizure. Seriously, this is less intuitive than the archaic text interface of the first game.
So I'm just sort of screwing around right now, not really clear who these people are or what Fountain Head is or why they're here as opposed to any other place on Terra. I'm just using the default party, and walked around town for awhile. I deposited all my cash in the bank, then found a sorcery guild apprentice, went back and got 50 gold, bought a membership, and then found that all the guilds and stores don't accept checks, went back to the bank, and widthdrew everything. I explored every accessible square of the town, and then went outside and promptly got ambushed by a pack of orcs. Reload. Then I found that the lattice gates can be kicked down, so I do this to explore even more of the town. Behind one gate is a pair of moose rats who murder my entire party.
So, any tips for getting started in this game? I intend to review the manual to figure out what all these cryptic icons actually do, but I suspect that won't be enough.

In my opinion, you're better off heading outside to gain additional protection from the armor class well. You'll have to head northwest, but the outside opposition should be manageable. You can then take out some monster lairs if you choose. Another option is to use the magic portal in Fountain Head to collect some Ultimate Power Orbs from the Pyramids.

May be you can find some help here :
http://shrines.rpgclassics.com/pc/mm3/

How do you figure that? The original keyboard shortcuts still exist, but now you've also got icons you can click on ;)
I for information
R for rest
S for shoot (this one's new, couldn't attack from a distance before!)
M for magic
Q for quests
F1-F8 to bring up a member of your party which have their own set of icons you can click or use keyboard shortcuts

fwiw, despite all the icons, I still use the keyboard shortcuts as WizardStan said. Its just easier for me to tap a key than to move the mouse around for everything.
The one exception for me is Inventory Control. For that, I find pointing & clicking to be easier.

You can hire 2 NPCs to help you in the inn. Buy a bow (and maybe other useful equipment) at the shop and equip it.
Don't destroy the lattice gates before you've gained a few levels.
Save your game just before leaving town.
After leaving town don't move, just shoot the goblins, if they get into melee range attack. When you win and get gold save the game if everyone survived. Cast healing spells if needed.
Move one step forward and repeat. Move another step forward and repeat. Turn left and repeat.
Move forward. You'll find something which looks like a white wagon, destroy it to gain loot and XP. You can return to the town for training. Look at the automap when you get lost.
With a few extra hitpoints you can survive longer.

Ok, I think I'm starting to get the ropes of this game. So far I've explored A1 and A2, cleared Fountainhead, Baywatch, and The Ancient Temple of Moo. I stole some decent loot from Castle Whiteshield (including gold weapons and armor) but the dungeon is too hard. I've gotten Mountaineer and Path Finder, and now I'm going back to explore A1 and A2 more thoroughly. There are a couple of things bugging me, though.
My stuff keeps breaking, and fixing it is getting really expensive. Money in general seems harder to come by, since enemies don't respawn. I hesitate to cast spells with a gem cost for this reason.
Baywatch cavern was an utter pain to map. There are all these pits of crazy goo that I have to wade in at least once, if just to map that spot, and healing via magic (so they don't die) takes forever. Only First Aid restores a good amount of HP for your SP, but I have to cast it six times to revive an unconscious party member. I can see this getting annoying in future dungeons too.
And two of my characters seem to be artificially aged. I don't remember when this happened. How, and how can I reverse this?

Welcome to a new (in M&M) game mechanic. It happens when you're knocked unconscious...an item may break.
Do you have the Levitate spell yet? And when you say map, you are using the Auto-Map feature along with the Wizard Eye, right? Plus there's the Hint Book.

I see. I played 1 & 2 with the mindset that unconsciousness isn't a big deal since you can just restore health with magic. Time to break that mindset.
I have levitate. I still take damage (and go crazy) from the slime puddles. I'm not really using Wizard Eye much, just cartography.

You'll find a way to raise the Fountain of Youth later in the game, which does remove unnatural aging. I remember Ghosts being able to age characters if they touch them.

I guess your characters were artificially aged when they were damaged by the phantom in Baywatch cavern, there isn't another monster with aging effects when hit in the early areas.
There are also a few monsters in the game (like ogres) which can break armor on a successful hit. Don't wear expensive body armor, you loose a lot of money if it breaks, sell it. Money put at the bank increases extremely fast, so so put all the money you don't need there, especially before you do something time consuming, like training or visiting a temple.
To avoid taking damage from acid on the ground like in Baywatch cavern cast "protection from elements: acid". All acid damage you take is reduced by 5+2*caster level (often to 0) until the next morning. The power shield spell reduces all damage taken by the caster level.
Gems are beneficial but you can afford to waste thousands of them during the game. There are methods to farm gold, gems and xp in the game, but you really don't need them.

Does money grow in the bank in all Might and Magic games?

In Might & Magic 3 plus Worlds of Xeen, you do get interest on money you deposit in a bank. For MM6, the bank is just a collection point that provides a safeguard if you're doing an Ironman run.