At the start of the second week hire a hero, buy him some creatures, and send him along Path #4. There are a few mines in this area along with a school, a wandering teacher, and some artifacts that are guarded by creatures. This area isn't essential but it does provide some good resources and a gold mine. Artifacts don't carry over to the next scenario, so it might not be worth the trouble to get some of them. I would definitely flag the gold mine at the very least.
Also at the start of the second week, Christian should have gone to some places of learning along Path #4 and then moved on to Path #5 to take the first castle. There are two enemy castles in this area (O1 and O2), but they are easily overtaken, especially if the hero who explored Path #3 comes back with some artifacts for Christian. Hire troops in O1, give them to Christian, and continue on Path #5 to take over O2 and any enemy heroes along the way. Leave a hero in each of the towns to defend while building and upgrading so you will have access to more troops.
If you see a hero, right-click or press option over them to get an idea of their army size, or use the Vision spell if you have it. If there are a lot of troops, stay in the castle because they are likely to attack it. If you build a Citadel or Castle, you will probably win because of the protection the walls and towers provide. If the enemy hero has a small army, wait until they get close and attack with Christian instead. You don't want to leave any enemy heroes wandering around the area.
Castle O3 is the last enemy town on the surface. Give Christian as many troops as you can gather and send him along Path #6 to take it. If you played quickly, your hero and army should be more powerful than theirs. There is a hill fort northwest of O3 where you can upgrade any creatures that aren't upgraded yet. You are very close to winning now, and you control the entire surface. You may see a hero appear from either E1 or E2 with a sizeable force so be ready for that by continuing to build and hire troops in O1 and O2. Have Christian and another strong hero recruit creatures and try to stop anybody that comes to the surface.
If you want to "groom" the four heroes you can take to the next level, there are many places of learning along Path #7 including a colloseum and an acadamey. There is also a jail as indicated by the J on the map which contains a hero with a sizeable army who will join you out of gratitude. You can also continue building up your towns so you'll have a sizeable force when you head underground. I like building mage towers in each town to get access to as many different spells as possible. Make your own decision about how much time, if any, you want to spend making your heroes stronger. Doing this will make future maps easier but if you like a challenge you can always head underground right after conquering the surface.