Your character initially appears in the Lands of Lord British just east of Britain and the The Castle of Lord British. You may want to head to town to buy equipment or food, give Lord British money to gain hit points, and otherwise begin customizing your character. You can Transact with the king, and he will tell you to find the The Grave of the Lost Soul. You won't be able to complete this quest until you have some means of traveling over water, and it is strictly optional.
At this point you're free to explore the continent. All locations in the Lands of Lord British are accessible on foot, with the exception of the two landmarks and the dungeon in the middle of the inner sea (Mondain's Gate to Hell). While the eight towns in the area all look different, they are for the most part identical. Not every town has every shop, nor do they all have ponds to Drop pence into, but the shops they do have contain the same goods. The exception is the weapon shops, of which there are two types. The nearest complementary weapon shop to Britain is in Paws to the northeast.
Your main goal is to complete the quest given to you by the king in the The Castle of the Lost King. The castle is northwest of Paws, on the north shore of the continent's inner sea. The king asks you to kill a Gelatinous Cube, a dangerous monster that can be found on Dungeon Levels 3–4 of any dungeon. You may have to travel the land gaining experience and money before you're tough enough to take on this challenge, but once you're ready, head to any dungeon in the land. The specific dungeon doesn't matter, as they are randomly generated at character creation. You can either explore the dungeon to find your way down, or make use of the Ladder Down and Ladder Up spells to quickly reach the appropriate floor. Once you have defeated a Gelatinous Cube, return to the The Castle of the Lost King. The king will give you the red gem and tell you that you will need all four gems to power the Time Machine. If you've heard enough hints at local taverns, you know you will need the Time Machine to defeat Mondain, so it's time to find the other kings with gems.
Those other kings are found on other continents, so you will need some means of traveling over water to reach them. Purchase a Raft, Frigate, or Aircar (if it is available) in a coastal town to begin your journey. Your required time in the Lands of Lord British is over.