You can, as long as the previous computer was not a Mac and your new computer is Windows, and as long as the iPod is not so old that it connects using FireWire (unless you new computer happens to have FireWire). Having or not having the old computer does not matter...
You cannot transfer song files from iPod to computer using iTunes (except for songs purchased from the iTunes Store). However, there are third-party methods and utilities that can transfer from iPod to computer. If you do a Google search on something like "ipod transfer," you should get some links.
Once the song files are on your new computer, add them to your iTunes library.
NOTE: Previous purchases made by you from the iTunes Store can also be re-downloaded at no cost from the iTunes Store Purchased screen.