Hi,
I am typing and sending this post on an Amiga connected by ADSL. If you have a router with Ethernet ports, OS3.9, and an Ethernet network card, then you got all you need. Otherwise, the necessary software can be donwloaded (besides 3.9).
You'll need a TCP/IP stack. I use Genesis as it came with OS9. I suggest that you install the stack before you install the network card (ethernet) drivers.
Then, since you'll certainly want to browse the internet, you'll need a web browser. OS3.9 comes with AWebII, although I prefer Ibrowse. OS3.9 also comes with a simple email program. If you don't have 3.9, then you can download those pieces of software (shareware).
I have been surfing the net with 3.1, but doing so on 3.9 has the advantage that the latter includes all the necessary software. I've been surfing with an X-Surf Zorro card, and now I use a PCI card via a Mediator board.
If you don't install the router yourself (eg the phone company does it for you), then make sure that you know the IP address of the router and the IP address(es) of your valid DNS server(s). You won't be able to connect without such information. The person that installs the router should have no problem in telling you those IP's.
Subsequently, just configure your TCP/IP stack, start it, start the browser... and ENJOY!
I recommend that you get hold of a graphics card. Browsing the net on AGA is simply not good enough. You'll soon want a minimum resolution of 800x600 and 24-bit color, and AGA is not up to that standard (besides being quite slow at just 640x400-8bit). A flicker-fixer with scandoubler won't be enough.
Once everything is up and running, you'll notice a great deal of improvement compared to dial-up.
Regards.