First off, our modern idea of "quick" and a person in mid-19th Century was completely different. In fact, in 1851 there was no real such thing as "quick"

Using your specific example, telegraphs were being adapted and spreading rapidly especially in the populated areas. However, they weren't phones. They were relayed messages, and the longer you sent one, the more the relays. It's possible that a telegraph could go from Virginia to Memphis in 1851, but it's highly doubtful in my mind. (you might have sent a telegraph to Washington DC but probably not to Memphis)

There was no real long range private courier service like we have now. The "American Letter Mail Company" was shut down by the US in 1851. Sending one person (a man in 1851) to Memphis would cost a fortune. He would have to travel most likely in two directions, either going North and Northwest to around Pittsburgh, and then by river or rail to Cincinnati and on to Cairo, Ill by river, and down to Memphis, or, down the James River to the coast and getting a boat to New Orleans (or multiple boats first to Charleston, SC, and then to New Orleans) and up to Memphis. This might have been more likely if you lived along an interior river (say St. Louis or New Orleans) and knew personally river boat captains, that went to Memphis regularly.

But, most likely you would send it via US Postal Service Mail. You could get a letter to Memphis, from Virginia. Stamps issued were only 4 years old, first issued in 1847. Mail was based on distance it had to travel. in 1851 the Cincinnati, Hamilton & Dayton Railroad had just opened in 1850.

In the end it could take months to get a message from rural Virginia to Memphis, TN, and you would have no way of knowing if it got there at all. The mentality of the person at home, would not even thing "I have to get an urgent message to a person in Memphis" . That's a 20 century thought. In the mid 1800's the thought would be, "I'll handle my issue here as best I can, as I'm here, and they are...God only knows, as I don't, by the time time they get the message, and could return it's months even a year and a fortune in traveling, so what could they do?"

Most people at this time, never traveled beyond the county they were born. Never traveled more than 10 miles away from home. Those that did travel, either did it for a living (shipping, trading, etc), or did it one way (Moving West) never to return, and in many cases never to hear about family again. They might get a letter that Mom, or Dad, or a sibling had died, but that's about the extent.

Letter prices set in July 1st, 1851: Up to 3000 miles, per ½ oz.: 3 cents. Collect: 5 cents. Over 3000 miles, per ½ oz.: 6 cents. Collect: 10 cents. Drop letters: 2 cents Source: http://www.tngenweb.org/tnletters/rates/