Good to know you're playing HoMM3- not only is it the best of the series (IMO), it's also the one I have the most experience with (so I can now provide you with quite a few additional tips).
With HoMM3 there are basically two ways to kick off the start of a game in the first week- build gold production or build creature production. Both methods have their merits (depending on the specific map), and knowing which one to go for can make a big difference in how well you start off the game. For building up gold production you'll want to produce Town Hall -> City Hall -> Fort -> Citadel -> Castle -> Capitol, basically ensuring you get the capitol built ASAP, then afterwards building up creature production mostly during the second week. For the other method, building up creature production, you'll want to ignore building gold-producing buildings for the first week and instead build the base creature production facilities (don't upgrade them yet), along with a Citadel (typically you'll want to hold off on a Castle until the end of the second week, as that 5000 gold is better spent on recruiting creatures at the start of week 2, or getting a Capitol built ASAP in the second week).
As for when to use these two approaches, if you're playing on the easier difficulties (which start you with lots of resources), and/or there are lots of free resources for you to grab then it's usually better to build up your creature-producing capabilities first (as you'll have plenty of gold to get you started out). Also, this approach is preferable on smaller maps, as having a decent-sized army ready to go at the start of the second week will let you take the fight to nearby opponents quickly (or just expand your territory and capture resources), and if you gain an early advantage on a small map it can be very difficult for opponents to claw their way back. On larger maps, and on maps where you start with fewer resources, it can be better to build up your gold-producing capabilities first, as it doesn't help having lots of troops to recruit if you have no money with which to recruit them.
Still on the topic of building up your town, try to get all basic creature production facilities built before using turns and resources to upgrade them. There can be exceptions to this, though, where an upgrade confers a very significant advantage that will help you clear neutral stacks much easier (and potentially earlier than you would be able to without the upgrade). Examples of such advantages would be picking up the ability for ranged attacks (gremlins -> master gremlins), gaining the no enemy retaliation ability (hellhounds -> cerberi), or picking up a second attack (archers -> marksmen, wolf riders -> wolf raiders). Also, while those level 7 creatures are often very tempting to try to buy early, they are very expensive for their capabilities and it's often better to hold off on buying them until you have a strong economy going. On that topic, though, the base level 7 creature dwellings are often reasonably cheap to produce, it's just upgrading them and buying the creatures that will break the bank. So get the base dwelling built and accumulating creatures, then once you have plenty of gold coming in upgrade the dwelling and buy the creatures that have been accumulating.
The stuff I mentioned about quickly recruiting a second hero for scouting and about using additional heroes to keep supply lines running still holds in HoMM3.
For developing your heroes, I'd consider Logistics, Wisdom, and Earth Magic to be essential skills for any primary hero. Pathfinding can be incredibly useful or fairly useless depending on what kind of terrain the map you're on has, so make a judgment call for each individual map. Other useful skills are Offense and/or Archery (depending on whether you focus more on ranged units or melee units), Air Magic, and Water Magic. Armorer can be good if you're going to be duking it out in fairly evenly matched battles, but those are the kind of battles that are better avoiding through good strategy, so your your judgment on whether that skill will be useful to you. If the map involves a lot of sailing between islands then navigation is good to have, but for most maps it's useless. On larger maps that will last a long time Intelligence is a good skill to complement other magic skills, as it will give you a huge mana pool that can really make a difference in big drawn-out battles. The usefulness of Luck and Leadership depend on what kind of artifacts you're able to acquire. If you're able to boost your luck and leadership up to +3 with just artifacts then there's no point in taking these skills, while on smaller maps where you probably won't be able to put together a killer collection of artifacts these skills can be quite useful. Typically useless skills which should be avoided are Eagle Eye, Scholar, Learning, Scouting, First Aid, Artillery, Ballistics, Estates, and Sorcery. The usefulness of tactics and diplomacy depend heavily on the map and your play style.
For battle tactics, there are a couple of good tactics to be aware of. First off, get used to using the Blind spell, as it's incredibly useful for taking a powerful stack out of action. When fighting other heroes it's often a good idea to let them cast their spell for the round first, so that you can either counter what they do, or ensure that they don't have a spell for the round to counter whatever you do. If you're playing more defensively with ranged units then mass slow is an excellent spell, while if you're playing more offensively with melee units then mass haste combines very well with blitz tactics. If you grabbed Expert Water Magic then the Forgetfulness spell is excellent against enemy archers (it completely negates their ability to use ranged attacks). If an enemy stack contains powerful melee units then you can make those units work for you with Berserk, as it will cause a stack to attack the stack nearest to it, even if that stack is friendly; try to time the casting of this spell just before the targeted stack moves to ensure that the spell can't be dispelled and that enemy stacks can't be moved away from the berserked unit. Finally, a cheesy (but effective) tactic for minimizing losses in battles is that once the battle is almost won, blind the last 1 or 2 enemy stacks, then have all your units skip their turns while you resurrect any fallen creatures (you'll need Advanced/Expert Earth Magic for this). With a high spellpower and knowledge you can bring back thousands of creatures while the opponent can do absolutely nothing about it.
Also, some a final starting tip about town selection for beginning players. The Castle town is probably the most balanced and easiest to play town for a beginning, while the Inferno is the most difficult. Stronghold and Dungeon are also pretty easy to just get into and play, while the Tower, Fortress, Rampart, and Necropolis can be a bit more challenging to figure out the best tactics for.