To go from coaxial (the BNC connector, 10BASE2) to twisted-pair (the RJ45 connector, 10BASE-T or 100BASE-T on Amigas), you'll need a media converter, and they can be almost as expensive as a new Zorro Ethernet card. You may want to compare the cost of media converters against the cost of two new X-Surf Ethernet cards from a vendor like AmigaKit, especially since it sounds like you only have one network interface card.
Once you have the proper equipment, your best options for TCP/IP connectivity are AmiTCP 3.0b2 (free) and Genesis (bundled with AmigaOS 3.9). There are many software options available for sharing data between Amigas and other systems, including standard file sharing protocols like NFS and SMB and commercial products like Amiga Explorer.