When buying equipment, buy armor first. Weapons don't increase your damage as much as armor increases your defense. Also, this will keep you alive much longer to actually deal out damage to the enemy.
Dragon Warrior 3 is an old style RPG. That means that levels are the most important thing. If things start to get difficult, take some time to level up.
Parry and Attack! This helps increase your defense. If you select Parry, then cancel it and select attack, you get the defensive bonus of Parry without wasting a turn. This is very helpful with bosses.
Use all seeds and Lifeacorns on the Hero. The other characters can become other classes to raise specific areas. Their stats are also halved when they change class, so you'll lose some of the effects.
Check all vases, sacks, dressers, etc. These will sometimes contain items that will be very helpful on your journey, such as seeds.