You can go online using the Amiga's own serial port. All you will need is a TCP/IP stack which will be either Genesis, which is to be found on the OS3.9 CD or Miami which you may struggle to find a registered version of.
You should find that the software has the appropriate modem drivers.
In the long term, you might want to consider a serial port expansion (Silver Surfer, Twister, Port Jnr) as the bog standard Amiga serial port is driven by the Paula chip (sound chip) which tends to run out of horespower quite quickly.
