I've heard a lot of Loyalists fled to the UK during/after the war, but how common was it for non-loyalist Americans to go to the UK, or for Britons to travel to the USA in the decades after independence? Did they have travel restrictions, or requirements on what they could do, what business they could conduct, etc?
By travel, it sounds like you mean movement without immigration, so I'll proceed on that premise.
Class has much to do with this question. For people with money enough to afford the passage, it was often very easy to travel between the two nations. In fact, for many American elites, a trip to London was a badge of gentility, which allowed them to display their worldliness. Likewise, English nobles and elites sometimes explored the United States after the Revolution. Rather than using this as an opportunity to improve their cosmopolitan affectations, however, many of these Englishmen wrote travelogues denigrating the culture of America. In the first half of the nineteenth century, British visitors to the U.S. wrote about 150 accounts of their travels that circulated both in London and in the United States (Sam Haynes, Unfinished Revolution, 24). This sort of traffic (more like a vacation than anything else) was socially limited, though relatively common among wealthy Britons and Americans.
On a larger scale, people with a good reason to visit the other state after the American Revolution usually would have had no trouble doing so. The Jay Treaty, which took effect in early 1796, normalized economic relations between the two countries until the French Revolutionary and Napoleonic Wars created havoc in the trade patterns of the early Atlantic. During this period, and particularly in the lead-up to the War of 1812, traveling on the Atlantic was difficult and risky for Americans. War between France and Britain (basically between 1792-1815) gave each nation a strategic interest in obstructing the other nation's commerce. As a result, France's naval forces sometimes stopped or even attacked British or American ships moving between the two nations. These wars drained Britain's manpower, and they became desperate for sailors. As a result, the British navy sometimes impressed American sailors (esp. in the years preceding the War of 1812). These issues certainly made transatlantic crossings much more difficult for about two decades.
Sources:
P.J. Marshall, Remaking the British Atlantic (2012)
Sam Haynes, Unfinished Revolution (2010)