It's very easy, I promise.
First, install Lightroom on your new computer. You can download 1.2 and just input your serial number instead of having to find original discs.
Then, on your old computer, find your current catalog. The easiest way is to open LR and go to File > Catalog Settings. That will tell you where you've stored it.
You want to copy all of your photographs, and that catalog file, onto your new computer. You could put them on a dvd and transfer them, or use a usb flash drive, external drive, or data cable - anything along those lines. Don't delete it off the old computer until you know your new one's all set up and working.
Now, on your new computer, hold down Alt while you open LR. That will bring up a box asking you where to find the catalog. Point it to the catalog you've just copied over to your new computer.
Hey presto, it should open your catalog along with all the settings.
If it can't find the pictures, the folder names might go red, and there might be a little question mark on the top right corner of each image. Right click on the red folder name, or click on a question mark, and tell it where the photographs are now.