A few definitions are required...
"with upgrades" - do you mean only upgrades that substantially keep the computer the same "basic computer" as it started, or do "rip the guts out and replace them" upgrades count? A Color Classic with PPC upgrade would be a usable surfing machine. As would an original Macintosh with its guts ripped out and replaced with a modern small-form-factor PC (like a Mac Mini.)
"surf" - do you mean use a web browsing program natively on the OS, or is simply finding a way to display websites on the internal screen sufficient? I have gotten my original Macintosh to "surf" by using a serial terminal program, and connecting to my Linux server. Then using a text-based command line browser.
I would say the oldest "usable" machine for browsing would be the Macintosh IIx. With a 68030, it can run iCab 2.9.9, which is still reasonably useful. Or, for a more "classic" experience, an SE/30. Slightly newer, but as a compact Mac, it gives a more vintage feel. (I used an SE/30 for web browsing for quite a while after it was "obsolete", as it fit in a cubby in my kitchen perfectly.) And, of course, you can get an Ethernet card for the SE/30 just fine.