You can have many possible configurations for connecting to the Internet with the Slackware Distro. The most essential part is the computer, but the most basic part is the most current Slackware Distro CDs. You can download these via FTP or HTTP, and rsync is the latest way to acquire the distros, but we won't do it that way seeing that we are just starting out here.
You should have one of two ways available to get into the Internet. It is somewhat foolish to make a guide on how to get online with Slack if we aren't assuming you have another way to connect to the Internet with another (or the same) computer. We will basically copy the way your current PC connects to the Internet with it's respective Operating System (OS).
The first way is via Ethernet. This uses the 10/100/1000base-T hardware, and TCP/IP for data packets.
The second way is via Wireless. This uses the 802.11 hardware, and TCP/IP for data packets.
The third and least common today is via analog phone lines using a modem to modulate digital bits to analog datastreams. This uses the modem as the hardware, and TCP/IP for data packets linked with the PPP connection protocol.
The other ways include variations of these protocols and hardware. Such as a Cable/DSL modem connected through 10base-T ethernet to a consumer router/firewall, and translated with NAT to either/both 802.11 and/or ethernet. What this reduces down to is that the direct link from your computer to the closest Internet related device will be, ~98% of the time, an ethernet cable or an 802.11 wireless uplink.
Also included in various hardware connections are IrDA, Bluetooth, Firewire (IEEE 1394/iLINK), and several others. These are not any more "advanced," but are simply not main-stream.