I need to upgrade from dial-up to a broadband connection.

To upgrade. Think of up + grade (level.) to upgrade is to go up to a higher level, to improve. If you upgraded your computer you might sell your slow, old one and buy a new iMac, for example. Dial-up. The old fashioned type of Internet connection which uses a modem and a telephone line - typically a very slow connection. Broadband. A more modern type of connection to the Internet which is much faster than dial-up. (broad means wide.) This sentence is an example of computers, technology.