Hello!
In the mid '90s, until I got an Amiga 4000 later on, I used to connect to the Internet with my A500+ with an Oktagon 508 SCSI controller and memory expansion.
I had, beside 2MB Chip RAM, 2MB Fast RAM: pretty much sufficient at those times (the worst thing was the lack of decent number of colors for iBrowse).
I used AmiTCP (from Aminet - the "previous version" of Genesis, on 3.9), a program for dial up modem (don't remember the name - try search on Aminet "dial up"), and the classic clients for WWW, e-mail, etc.
Bear in mind you have to cope with scripts to configure AmiTCP and the dial up program properly.
Later I used Miami (shareware version always from Aminet), but that depends on your system resources, particulary Fast RAM (I think having 2MB at least for a minimum decent way to connect).
BY!