I would like view a list of who logged on to my server.
You can also use the 'w' command.
You can look at network connections using the 'netstat' command.
pts refers to pseudo terminals (i.e. not hardware). In your listing they are probably shells opened in a GUI.
tty1 is the first terminal. You probably also have tty2 to tty6. If you use Alt-F[2-6] in tty1, you will probably get tty[2-6].
down means that system has gone down, probably for reboot.
The +1 means to add a day i.e. 24hours.
root tty1 Thu Mar 27 00:04 - 00:28 (1+00:24)
So you had a root shell open on the primary terminal for over 24 hours. Not a good practice from a security point of view.