Welcome to linux.
There are numerous ways to connect your pc to a LAN that provides it with internet access. Perhaps you could make it easier on us to answere your question by providing more information.
1. Does your home lan have any kind of wireless security enabled such as WEP/WPA or MAC address filterting?
2. What have you tried already?
3. Are you saying slackware was your first linux distro or is that what you are currently using. I feel obligated to remind you that this forum is dedicated to users of the BackTrack Linux operating system and questions regarding other distros will most-likley not be met with pleasent responses.
All that being said here are a few things to try.
fire up a command prompt shell and type 'iwconfig' to make sure your os recognizes you have a wireless card. If it does try typing 'dhclient interface name' (i.e. 'dhclient ath0') or 'dhcpcd interface name'. your interface name will be displayed from the 'iwconfig' command.
if you have some type of encryption enabled on your router you will need to specify that with the iwconfig command. For example if you're wireless router was name 'wlink' and your WEP key was '12345678910' and your wireless card was named 'ath0' than you would first type.
iwconfig ath0 essid wlink key 12345678910
If done propperly that should set up a dhcp enabled connection between you and your home router giving you an IP address on your local subnet and enableing your PC to recieve access to the net.