You wouldn't be transferring from bearshare to bearshare, you know that right?
bearshare is just a program used to download and upload files (typically illegally, and we don't allow any talk of that on this forum apparently)
Just go to the folder all ur downloads go, copy them to a flash drive (or DVD, CD, HDD, cloud storage, network drive, or any other form of removable storage), then copy them some where onto the new laptop from the removable storage.
But, scan the entire folder for malware/viruses beforehand like COLGeek said