Before you read the spoilers, you could also check my strategy guide for Heroes I campaigns. With good use of this general guide, you migth be able to complete all campaigns without spoilers or walkthroughs.
For more detailed campaign descriptions, alternative (at some places very different to my advice) strategies and about Homm I in general, please see Hommworld
The easiest way to complete this map is to go straight for the central town. The one dragon guardian is not stronger than the main heroes/armies of your opponents. If not even during first week, by the begin of second week this castle should fall easily.
If you have any problems on this map, you are spending too much time on exploring the wilderness or are “preparing for war” too long. I would suggest using teleporters and not to mess with boats when travelling around
You do not need to visit all obelisks to find the “eye of Goros”. After the area on puzzle map is recognisable you may command heroes to dig up the area step by step. Eventually you find the grail. Most likely you need to conquer opponents first and then search for grail. Note that you need to remember by yourself, which obelisks are already visited. User interface doesn't help you here
This is a though map and there is no way to give exact recipe how do proceed. The factions are located: Warlock: SW, Barbarian NW, Sorceress NE and Knight SE. You better strike your first opponent fast and after conquering him use two “main” heroes, since one cannot be handle all opponents on a such open map.
War against Lord Slayer
This map may be the most challenging in entire campaign. You may have a barbarian hero attacking your castles with 7 cyclopes in about 10 days after map begins. The first hero is followed by a constant flow of attacks. If the challenge is too hard for you, restaring the map helps sometimes (i.e. the barbarians are not always very aggressive). Build up both of your starting towns in order to have as many creatures as possible. With warlocks or sorceress, hire a might hero to fight. Do not venture far from your towns with the main hero, or use chaining. Letting AI to take one of your towns causes him to divide his army, thus making it an easier prey. Sometimes it may help you. After time passes you grow stronger and can take incoming barbarians easily. Moreover, the AI cannot send strong armies endlessly, at one moment his supplies are depleted. Now it is time to visit their homes. Do not bother with the mines in deep desert; go straight to the passage (NNW from your starting towns). About 4 days travel to East from the passage takes you to the main town. The hero here has a very strong army, but he won’t leave his castle. To win you need to conquer him
War against Queen Lamanda
This is a relatively easy map and you can feel safe in your starting area. To go to the Sorceress lands you need to reach first a portal and find a village on the other island. Build shipyard and ship there. The best is to hire a sorceress and let her sail to the coast of your main island. Now board your main army and go North, the passage lays in the middle of the map. Sorceress cities are in deep forest. Two, most separated ones are in W and NW from landing area. Heading to one of the East roads takes you to several towns, and reaches also the final castle.
To find the ultimate artifact you need dimension door.
War against Lord Ironfist
You are not completely safe on your starting island and sooner or later the knights invade you through the portal. To reach the final castle in NW area you need to conquer a knight town on river shore and build a boat. Most knight towns are close to each other making it easy for them to recapture anything that you leave unprotected. Consider splitting you main army into two (or creating a new strong army) to hunt your opponents down.
War against Lord Alamar
Once again your starting area is quite safe. The main hint is not to enter the maze with the aim to travel through it. Only interesting thing in the maze are mines. Build mage guilds up, and use dimension door to reach warlocks. The main warlock town that you must conquer in order to win is in the corner.
The final map! In contrary to most previous maps there is few free resource piles lying around your town. The dragons city that you need to conquer has lots of dragons (plus a horde of griffins), thus, you most likely need to conquer all opposition before you can safely handle the main goal. Location of factions is: warlock SE, knight NE, sorceress SW and barbarian NW. Again, sticking fast to the hearth of opponents (starting town) and not spending too much time on “preparations” and exploring wilderness may be the key.