Start LimeWire and wait for it to connect to the network. You should see 4 green or blue bars in the bottom left corner. Once connected, search for the songs you want. Then download them.
To download files:
Double click on the file you want or right click it and select download. The files should start downloading in the download window, below the search window. The files will be saved in the LW Library. By default it is located at, C:\Documents and Settings\"your account name'\Shared(Just go to the Library page in LW).
If the file doesn't show up in the download window, see this thread:
"License requirement Option".
There are also a lot of fake/spam files that'll act like they're trying to d/load but never will. So be patient and keep trying until you find what works. This thread has all the tips for d/loading in it:
How to find music.
If you can't connect then you'll need to post the details of your set-up:
System details - help us to help you