There was only one way to communicate with other computers and that was with a modem and telephone. You could only call one number at a time, so you could only talk to one computer. Later the telephone company perfected multi line capabilities and many people could call one computer and talk to each other, but to call another city cost long distance charges. (friend of mine kept calling someone in California and ended up owing the phone company a couple of thousand dollars.)
1. Home computer users mostly bought programs to load onto their computers to run. A serious home application at that time would have been a cross indexed list of books in ones private library, recipes, or balancing the check book.
2. If you wanted to surf you got something like a Hayes modem and had it connect to a bulletin board. Bulletin boards were where people posted stuff for others to read, look at, copy or download.
3. If you could afford AOL after it stopped just being a place to rent games, you could use tools like gopher, webcrawler, telnet, and ftp.
4. If you happened to work for a big company or institution they would might of had a minicomputer or maybe even a mainframe. These would be connected to different offices via cables (amazing, just like today). Many places also had the ability for employees to dial into the computer. Security at the time was very lax, and the software often did not disconnect a user if the phone line lost connection, so it was common to dial in and discover you were already logged on but as someone else and could see what ever that person could see or run their programs.
5. Watch the movie War Games with Matthew Broderick and Ally Sheedy. It will give you an idea of what the home and institutional capabilities were at the time (No the computer center about to blow up the world did not exist, at least as that fancy). It is from 1983. The home computer capabilities it shows actually date from the late 1970's.