I'm new to the game. For example, is there a way I could start off as Dace for my first Hero when I start a new single player game on a random map? If so, how? Thanks.

After selecting the random map settings, bring up the pane with player color selection. There is an option for choosing your starting town type, selecting your starting hero (assuming you chose a town type), and your starting bonus (artifact, gold, or resources).

Thanks. So I can choose Dace the Minotaur Overlord doing this ?

Yes.

Note that this also work on non-random maps, unless the map maker specified other starting hero(s).

On single player campaigns though, your starting heroes are predetermined.

If the map settings don't require a specific town and/or hero, then you may choose your town. You may only choose a starting hero that is associated with that town (no picking Solmyr as your starter hero for a Necromancer town), but otherwise you can pick a starting hero as well. Random maps don't have the town/hero pre-set, so you can choose while setting up map options.

Thanks guys!

Man this game is awesome!

How do HOMM1 and 2 hold up?

HoMM1 is quite rough by comparison. Later games in the series smoothed out a lot of things. It is still playable, though. Biggest change is the move from individual instances of spells to spell points. In HoMM1, your hero could visit a Mage guild, and he would learn one instance of each spell in the guild per point of knowledge; for example, 5 wisdom would let him cast 5 lightning bolts before running out.

HoMM2 holds up a lot better. Personally, I like 3 more, but plenty of people like 2 better. 2 introduced spell points, hero skills, and having to build town defenses (like towers), among other things.

Just out of curiosity, why do you like Dace in particular?

Re HoMM1 and 2: I usually advice people to play all games in a series in order, even if it means playing "worse" games first. But HoMM1 is an exception. If you're new to the series it might be tough to get into.

If you like HoMM3, give HoMM2 a try; it's an excellent game. But only check out HoMM1 after you become more comfortable with the series. It's not newb-friendly.