Several hints which might help:
I've just bought a new USB mobile broadband dongle, which on the device has an install script and a few other files with it too.
If you include the make and model in your post, some people might have more specific suggestions or might have already managed to get the same device working.
Most broadband modems I've seen have two working modes: mass storage mode, and serial modem mode. They tend to switch between the two. Last time I looked, the package usb-mode-switch (the exact spelling seems to vary) handles this. If I remember correctly, it installs a bunch of udev rules, which recognise various usb 3G modems and handle them, forcing them to switch to the modem/serial mode when needed. I can't remember thought if it is included in Slackware by default or it needs to be installed. Checking their home page, it seems development is ongoing with recent versions. You need a recent version, as you need a recent version of their devices database which will recognise your modem.
In order to get anywhere (with ppp or wvdial or any other similar software), your modem has to be seen first as /dev/ttyUSBx or /dev/ttyACMx (if we are talking about a USB dongle). If, after waiting 1-2 minutes for it to initialise, those devices are not showing up, it probably means it is not switching from mass storage mode to modem mode. You can try watching dmesg after plugging the modem in, as many times relevant messages show up there in terms of what is detected.
After you solve the above problem, then you can proceed to configure kppp, pppd, wvdial or any other package of your preference to send the right strings to the modem for initialization and dialing.
Hope the above helps some