There are two different ways to initialize your modem under Linux.
Since you are probably using the graphical interface that came with your distribution of Linux, the first tools I'll be explaining at the ones that you'll use with your mouse. There are various tools available but this HOWTO will be explaining the two most popular ones that come with GNOME and Redhat 6.
modem-lights-applet
The modem lights applet is a really easy way to get online once you have everything setup correctly. You'll have to add it to your GNOME Panel by right-clicking on the Panel, going up to Add Applet and choosing the Modem Lights applet under the Networking menu. A small box with two dark green circles, a box and a red button at the bottom should appear on your Panel.
The bottom red button is the button you'll click to log onto SpinnNet. When you push it, a box will open that asks if you want to connect and when you push 'Yes' it should start dialing. The square is the transfer graph, it shows the data you are sending out to the net in green and the data that is coming into your computer as red. The two buttons on top are input/output indicators that tell you when data is coming in or going out of your modem. They are used to make sure your connection is working. If the lights aren't flashing, no data is moving through your modem.
gnome-ppp
GnomePPP is the default PPP program for GNOME. It takes care of a few things the modem applet doesn't but it is also a little more buggy and a tad harder to use.
more to come...
Dialing it at the command line is easy but you don't get any lights or anything to help you make sure you're connected. To dial SpinnNet with the chat script you created earlier, type the following command:
/sbin/ifup ppp0
To see what's going on with your connection, you have to view the system log:
tail -f /var/log/messages
This is will show you the messages that pppd (the ppp daemon for Linux) is sending your computer. You should see various dialing messages and eventually, if all goes right, you should see your IP address come up. When this happens, you're online. If you get errors and your modem hangs up, try to see what's wrong and check the HOWTOs listed in the Introduction. If your modem doesn't even start dialing, it's probably either an incorrect device setting or a Plug&Play error. If you hear dialing, then it's probably a chat script issue.
There is also a program called pppstats which will help you determine if you're online in case of problems. To use it just type:
pppstats
Read man pppstats for more info.
To end your connection, type:
/sbin/ifdown ppp0
You should hear the modem hanging up.