The video cable runs from the card to the monitor. Switch ports
It's pretty important to get all the ati software installed so, the fan controls on the card work properly. On some machines if you have had no extensions and stuff for ATI in there before, the card may have some odd behavior until you get the software installed. If your old card is still working it may make sense to install all the ATI Utilities, THEN swap the cards, bootup, and run the cache cleaner (as mentioned above for nVidia cards, and in the ATI Readme) to make the system load the
The ATI cards don't have an alarm on them like the nVidia cards, If you don't hook up the power you simply get no video, (no annoying alarm, just a black screen). If there's a power problem and you have an ATI card, hints indicating poor power will be things like the card running hot, strange freezes or other behaviors as mentioned above for the nVidia cards. If the card runs hot too often, you will hear the fan speed increase and decrease, or if things are really bad, the fan will run fast most all the time.
Here's a link to the ATI utilities: