Your basic options for upgrading from dial-up is either cable or DSL -- cable you'll only have one choice for provider (most likely). Cable is usually a bit faster and doesn't require a contract. DSL (in my experience) can have outages, requires a one-year contract, but is less expensive.
You may find there are different DSL providers (usually phone companies like AT&T;), but there's really not a big difference between them. Like the cable companies, they're all equally poor.