If the new computer and the new printer can both work with USB ports, to print without the USB to serial adapter, you can connect the printer directly to the PC using the USB ports. However, in this case you will need to visit the printer's manufacturer website and download the appropriate driver. Without those, you will not be able to establish a connection. Also on the site mentioned, you should be able to find tutorials or installation guides. Be aware to download the compatible drivers with your OS to avoid compatibility errors.