I'm taking some of my answer from an older post of mine you can see here.

I'm going to break your question down to address a few things first. Without the contextual information, my answer will be less clear.

I'm a person in a mid-size city

There really isn't such a thing yet. By 1790, there were three cities with populations larger than 15,000 (Boston, New York, Philadelphia). Most people, 95% according to the 1790 census, lived on farms, away from cities. People really lived in either cities (of various sizes between 10,000 - 15,000) and towns (maybe around 5,000). Baltimore, which we think of a city today, was actually known as "Baltimore Town" at this time, with a population of around 5,700 around the time of the American Revolution.

and I'm going to visit a friend on the other side of town. Would it be typical to carry my gun with me? If I did, would that alarm people? What if I was traveling further (like between two cities). Assume that I'm not affiliated with a militia or military.

Absolutely this would alarm people. Every American didn't own a firearm during this period. In fact, many didn't. Firearm ownership was about 33% around 1776. It greatly increased during the war, due to militia laws.(1) In fact, this was alarming to most early governments, which relied on militias to be of service in times of emergency. Saul Cornell, one of the nation's leading historians on the history of early firearm ownership and the Second Amendment actually wrote an article about how early regulation of firearms was to make sure people owned the right type of weapon in case a militia needed to be called up. Here he explains:

...most Americans would likely have shown up for active duty with fowling pieces, which were more like shot guns than muskets, because these were better suited for putting food on the table.

Also, walking across a town like Baltimore or a city like Philadelphia wasn't very dangerous. First, the cities were a fraction of the size of what they are today. Second, crime rates were overall low in most areas. When crime did happen, local sherifs would handle it, and if there were multiple perpetrators, they'd call up armed members of the populace to handle it.

1]Robert H. Churchhill. “Gun Ownership in Early America: A Survey of Manuscript Militia Returns” *The William and Mary Quarterly,*Vol. 60, No. 3 (Jul., 2003), 615-642