Powering with the barrel jack and connecting gia ethernet will work. By default, it assumes that on the other side of the ethernet cable there will be a DHCP server running, which is responsible for giving the board a balid IP address. The easiest way to do so is to connect it to a home router. The board will NOT have its canonical 192.168.6 or 192.168.7 addresses , but an IP in the DHCP's valid range. For home routers that's typically 192.168.0.x or 192.168.1.x or 10.0.0.x. You normally don't need to know the IP and can just access it via bela.local . If that doesn't work (e.g.: in the rare caee where yyou don't have a avahi/zeroconf/bonjour/mdns client running on your system), then accessing the router's admin page you should be able to see the actual IP.
Connecting directly to a host computer via ethernet, it will.most likely not work immediatelt, because you are unlikely to have a DHCP server running, so the board won't be assigned an IP address. The trick is often to enable internet sharing on the host, which should give the board an IP address. It will be hard to find out what address it has been given, but it should become accessible via bela.local