Hi Carlos,
The "real" start of the Internet for Macintosh computers was with the arrival of usable web browsers around 1994 and 1995. As has been indicated already, a typical way of accessing the Internet at that time was via a (dial-up) modem.
For a Macintosh SE (with an internal or external hard drive) or a Macintosh Plus (with an external hard drive), the following would allow dial-up Internet access:
1. 2.5 or 4 MB of RAM.
2. External serial modem. Almost any modem with a speed between 2.4 kbps and 56 kbps can be used (a "PC-style" modem with a DB-25F port would require a DB-25M to MiniDIN-8M Mac modem cable). Connect the modem to the Modem port of the computer.
3. System 7.0 (or higher).
4. MacTCP. This software handles the Internet protocol on the computer.
5. MacPPP 2.0.1 (http://archive.info-mac.org/_Communication/_Internet/_Connections/_PPP/).
6. A suitable web browser (such as MacWeb 1.00A3.2).
7. If necessary, an email program. This could be an early version of Eudora. However, please note that SMTP authentication required by many/most ISPs often prevents the use of an old email program.
8. If necessary, an FTP application (for example, Fetch).
For Ethernet-based Internet access, an Ethernet card could be used in a Macintosh SE. A Macintosh Plus would have to rely on an external solution (SCSI-to-Ethernet adapter).
BTW, with the correct software and hardware, it is even possible to set up a Macintosh SE or a Macintosh Plus as a web server.