What was a main issue that colonial America had with the British Empire?
The most commonly cited issue is often oversimplified in the phrase "no taxation without representation".
Ostensibly, the colonists were Englishmen (citizens) and therefore entitled to certain rights and protections under the terms of the English governing laws. However in practice this was not always the case, and the colonists came to see themselves less as brothers and cousins of their European relatives, and more as servants and slaves of a corrupt and unsustainable system that sought to exploit them.
A primary antagonist was the Seven Year's War, which unsettled the British economy and altered the political landscape in North America. Not only were colonists irritated by the idea that they had to help pay for a war they were not really a part of, but the British government began exercising various taxes and trade modifications that the colonists were not consulted on, and which were not helpful to them. Additionally, the British government made several geographical rulings that prevented American colonists from expanding westward, into Indian or former French territories.
In response to these protests, the British government continued to enact legislation that further punished or restricted the colonies, leading to the growing sentiment that the colonists were not seen as English citizens and therefore had no reason to respect the authority of the empire.
Taxation without representation is, of course, the best known, especially with the Stamp Act and Townsend Act. The British were trying to recoup money lost during the French and Indian War, so they passed the burden onto the colonists with those two acts. Other than that...
You've got the Proclamation of 1763 where the British Empire, in an attempt to control expansion, forbade anyone from going past the Appalachian Mountains. This meant that anyone who already lived there had to uproot themselves and move to an "approved" area.
Then you've got the Quartering Act where American colonists were required by law to not only allow British soldiers to sleep in their homes, but they also had to feed them. If they couldn't house them, they had to pay for their lodging elsewhere, even when there were no wars. This developed into our Third Amendment, the statement that Americans would be protected against unauthorized use of their own property, that people could no longer be forced to house or quarter soldiers.
Definitely rampant and what the colonists believed to be illegal taxation by the British. It seemed that every time the British got into some skirmish they would turn to the American colonists to help refill the royal coffers, and they were getting sick and tired of it. Then, when they tried to voice their opinions, they were punished for it, in direct violation of the English Bill of Rights and the Magna Carta. Since the colonists had already been accustomed to virtual self-rule, it was only a matter of time that they would say enough was enough and send an ultimatum to Britain, which they did with the Declaration of Independence.
The colonists felt that they were unfairly taxed and treated by England. For example, the Quartering Act of 1765 required colonists to accommodate British soldiers in America by finding lodging for them. This was unnecessary, as with the end of the French and Indian war, there was no need for soldiers in the first place. Speaking of war, the parliament decided to impose taxes on the colonists to help pay for the French and Indian war. The Townsend Act issued in 1767 forced colonists to pay taxes on British imported goods. In 1764, the Currency Act eliminated paper currency, and the Stamp Act of 1765 required colonists to buy stamps to use on documents.