The goal of Spaceward Ho! is to conquer the galaxy. You start out on a single, wonderful, heavily populated planet. You have just become capable of building your first interstellar space ships. So you build a couple of exploratory ships and send them to nearby stars. Some star systems will be more habitable than others.
Once you find a planet that is similar to your home planet, you build a Colony Ship and go colonize it. Once there, you make its temperature better by terraforming it, mine all the metal from it, and then use it as a base for further exploration and colonization. After several centuries the new planet's population will grow so large that it will start making a significant profit, just like your home planet.
You'll hop from planet to planet expanding your galactic empire. Eventually, you'll run into opposition -- bad guys who are out to conquer the galaxy, too. Besides exploring and colonizing, you'll need to defend yourself against the more aggressive of these alien races. You can do this by building defensive satellites at your colonies. You also can defend yourself by building fleets of fighters and preemptively counterattacking the aliens -- the old "best defense is a good offense" strategy. If fighting isn't to your taste, you can befriend some of these aliens, creating allies instead of enemies.
In addition to building ships and terraforming and mining planets, you'll spend money on technology research. The first ships you build can travel to other stars, but they're big, slow, clunky, and weak. When you spend money on research, your ship technology levels will improve. To no one's surprise, better ships will destroy obsolete ones in battle.
When you kill all enemy colonies, you win. You're free to keep playing, but there won't be any resistance from the bad guys, since they're dead.