The official strategy guide to HoMM3 from Prima states there are four types of heroes: garrison commander, general, scout, and troop transport. With this in mind, and knowing that you have up to eight heroes on the map at any one time (any others must stay in town garrisons), I aim to be able to work you into playing multiple heroes without neglecting any.
The general is the kind of hero you've always been playing, and the one most refer to as "their main hero, or their combat hero". You want, among other things, to identify which computer heroes are their "primary generals", because these will be getting the highest level, and the strongest armies. If you can take them out, without them retreating, you will deal them a big blow, especially from the mid-point in the game to the later-stages after they've had time to advance (and taking potential experience such as from treasure chests off the map).
Say that in every scenario, the first hero you get is to be your primary general. You can explore the territory right around your town first, acquiring any loose resources within easy walking distance. Inside the town, build a town hall on the first day and recruit a second hero, preferrably one from your town side also, but mainly looking for one with the scouting skill already learned. Buy whatever troops you can now, and go to the first hero outside. Trade to the new hero, who will be your primary scout, all but one of the weakest troops from your first hero, and then this force will be your scout's army. If you have the money, buy a second new hero, and give all his troops to the scout too. This third hero will be your primary troop transport.
Most would advise giving the troops to the general, and the official strategy guide (OSG) says that, but in my opinion this overlooks something obvious: as any hero goes out into the black areas of the map, revealing it, he or she is scouting, so why not send your scout out first instead of leaving the general to do this? Having scouting (the skill) will pay off, and it can only do so once because the map will stay revealed. Now let me give missions to the three heroes, and building advice for the first ten days of the game.
The first mission of the scout is to locate mines, and the first of these being wood and ore if they are not already in view of your town. Any mines you can claim with him, go ahead and do it, even if you will lose some men fighting. If you can find a gold mine, it would be particularly useful to take it, and you should make a note of what guards it, and as soon as you have the armies to beat them you should. Usually, there will be one mine of each type somewhere around your town, and once you find them all (with gold mines being the rarest), if you could draw lines between them, encircling your town, you can mark this as your starting territory. The first priority of your war effort will be securing this, including taking all the mines.
Depending on what side you're playing, you may need some resources more than others, so keep an eye out for them. The rampart, for example, needs a lot of wood for the elves, and crystals and gems for pegasi, unicorns, and dragons, respectively. You should give priority to claiming sawmills, crystal mines, and gem ponds therefore. In particular, try to notice any loose wood, crystal, or gems and take these. When you play at the Easy or Normal difficulties, they give you plenty of resources to start with, though, so if you want you can bypass this mission with your scout. First let me say what the other heroes should do.
If the beginning will be all about getting the local mines, you need the right troops to take out their guards, and any other monsters in the way, along roads or what-not. Anything your scout with his army can't take out, make not of as a battle for your general to. Move your general adjacent to those enemies, and build creature buildings at home. Any loose resources your general can get, have him get, and get any chests for experience and any buildings for skill points or experience. The troop transport will then recruit the troops, and when you have enough, march them to where your general is waiting. After you win the battle, move on to the next battle and so on until all the mines are claimed. At the beginning of the second week, you'll have reinforcements, which the troop transport will again be able to bring to you, and this should be enough except maybe not for a gold mine which tends to be well-guarded.
The second mission of the game, as I play, encompasses the next ten days again. In this stage, your scout's mission is to find a new neutral town if there is one nearby. You may be able to take it without reinforcement, especially if you've been able to add to your army by visiting any outside creature dwellings, but the best practice is to just scout the area of the new town and leave the new town to your general to take. The time you take a second town for the first time is when you first need a garrison commander at the main town, by the way. Let's say you recruit one now, and everything is going well.
Have the garrison commander stay near the main town, visiting local creature dwellings to build up your forces and any buildings that offer your hero anything good. Work on building your capitol, and your mage guild, to which all your heroes should visit to learn all the spells he can. The scholar skill can be very helpful at bringing the spells to your more distant heroes, and this can be a great skill for a troop transport to learn.
The next stage of the game is searching for your nearest enemies. With the scout, you want to find your "borders" with them. Remember that borders are sometimes defined by mines, so if you come upon a mine he has flagged, there you go. You can flag it yourself if it's unguarded, and move along. If you keep your general near, the computer is less likely to attack your scout, and ideally your troop transport will keep reinforced your general and your scout to a lesser extent.
So in summary, every time you get a new town, you recruit a garrison commander for the last one you had. You recruit a new scout every time you need to explore in multiple directions over significant distances, and recruit a new troop transport for every new front. You recruit a new general for every new region you want to invade as opposed to just defend, if there are multiple fronts, and you use your main general to seek out the hardest battles on the map and to take out the computer's main generals.