It depends on your internet provider, and the type of internet being provided. Cable hi-speed internet (Comcast, Cox, and others) requires a cable modem. DSL high speed internet (AT&T) requires a DSL modem.
Yes, you will need a DSL modem, if you have a DSL subscription, to get broadband internet access. There are also other broadband internet access available though, aside from DSL.
Yes, the speeds offered by broadband far outclass anything a 56k modem can offer. Cable modems are generally faster than DSL, but even DSL blows dialup out of the water.
Actually, you don't! Using your Cable Company's modem / internet access is a LOT faster than DSL. I have Road Runner Turbo and it is 10 times faster than DSL internet!