The British Empire began to grow through colonization beginning in the early 1600s and even had control over 458 million people in the mid 20th century. Since the empire was so vast and held sway over so many colonies, below are a few ideas to help get you started concerning names of areas colonized and the years the colonies were established.
While Christopher Columbus under Spanish support started exploring the North American region, specifically the islands off of North America, in 1492, it wasn't until the 1500s that England began exploring and claiming North America. Specifically, in 1578 Queen Elizabeth I decided to rub shoulders with Portugal and Spain in exploration and colonization. By 1583, Humphrey Gilbert set sail towards North America and landed on the island of Newfoundland, which he claimed for England. Gilbert's half-brother Walter Raliegh next landed in what is now North Carolina and claimed Roanoke for colonization, but the colon failed due want of supplies. It wasn't until the early 1600s, after Queen Elizabeth's reign had ended in 1603, that England's colonization began to take hold.
In 1606, King James I chartered the London Company, also called the Charter of the Virginia Company of London, for the purposes of colonizing North America. By 1607, the London Company landed on Cape Henry near Virginia Beach and established the Jamestown Settlement by the James River. Later that same year, the London Company also established the Popham Colony in Maine. By 1612, the London Company's charter was amended to include more territory, enabling the company to also establish a colony in St. George's Town, Bermuda. The London Company's establishment of North American colonies was so successful that it lead to the further establishment of the Thirteen Colonies in North America, even though the company lost its charter by 1624 due to financial hardship.
In 1627, England also settled colonies in the Caribbean islands, then called the British West Indies. More specifically, in 1627, Barbados and the Leeward Islands were settled by William Courten. In 1632, the Islands of Antigua, Barbuda, and Montserrat were acquired. Other islands were also acquired, such as the Bahamas in 1647, Anguilla in 1650, Jamaica in 1655, the British Virgin Islands in 1666, the Cayman islands in 1670, as well as many, many more. After the Seven Years' War between England and France, new territories in the Windward Islands were acquired and colonized for England. In 1763, Barbados was placed in charge of the Windward Islands, which included Grenadines, St. Vincent, and Tobago. By 1802, England also acquired Trinidad as a colony and St. Lucia in 1814.
Meanwhile, by the early 1700s, the East India Company was chartered to expand British territories into Asia. By 1757, the East India Company had set up rule in Bengal and Bihar, India, after the Battle of Plassey. Expansion continued from there, and England had acquired Egypt by 1799, Java by 1811, Singapore by 1819, Malacca by 1824, and Burma by 1826.