|Windows 95 is far from dead!|
I doubt any ISPs actively support Windows 95, but if your ISP supports dial-up and TCP/IP then you're in luck and it's pretty much guaranteed to work on Windows 95 because Windows 95 supports both of these protocols.
I also agree with T-R-A that 3Com are the best network cards. Mine is a 3Com 509B-TCO ISA rinning in PnP mode. (This machine originally ran Windows 3.x which didn't support PnP so it was common to get ISA-compatible cards back then that could run in 3.x and NT 3.x)
For what it's worth I am typing this reply from my Gateway 2000 P5-133XL (133MHz Pentium, 128MB RAM, 12/27/1995).
I use a combination of Netscape Navigator v3.0 Professional (also, ftp here) and for newer sites that require SSL like eBay I use Mozilla Firefox 188.8.131.52 (available for download here. Run Firefox in Windows 95 :: Mozilla Stuff :: JohnHaller.com. Many of the links on his page don't work anymore, so before installing Firefox you need to install 50comupd.exe, dcom95.exe, dun14-95.exe, speu.exe, and winsock 2, all from Microsoft's website. **You need to install all these updates and reboot before you can install Firefox successfully**.
Also, the first time you run Firefox you will get a "invalid page fault". Just click close, and restart Firefox. This is because Windows 95 doesn't support Firefox's browser import settings functionality, but you will only receive this error once, when you first install Firefox. Restart Firefox and you are good to go!
For the majority of websites, I prefer using Navigator (as I am right now) because it runs much faster than Firefox on my old box, and loads all the websites I need to see with the exception of banking sites and eBay that require a new browser such as Firefox. After all I can login and post on computing.net using v3.0 so the SSL it has is compliant enough.
I also highly recommend ZoneAlarm 2.0 firewall or newer up to v 3.1.395. I think they all work the same and I have a hardware firewall anyway.
(Updated 04/14/10 16:17 with the link for Firefox in paragraph 5)