I have a Windows for Workgroups computer set up with TCP/IP and lots of networking software, so I can speak to this. First thing, Windows doesn't have built in TCP/IP. Back in the day folks used Trumpet with a modem, but people with Ethernet and the ability to run Windows for Workgroups should use the more modern and stable Microsoft TCP/IP stack. If you're trying to do this with Ethernet, I recommend you get WFW no matter what. Many of the remaining resources were written twenty years ago, when everyone had dialup, multimedia was by Real Player, and pages would render in IE 5.0 still.
For a browser, I highly recommend Opera 2.12. Opera was the fastest in the 90s, and this early version can load HTML 2.0 pages with acceptable speed on even a 386sx. It will run in 4MB of RAM, but 6 or more is better. Turn pictures off -- i know, i know -- but no 386 has the power to decode images on the Web and not grind to a halt. A 10KB banner GIF sure, but hit a 300KB JPEG and the machine will cry. Lynx was never pointed to Windows 3.11, but this will be rather like a mouse-centric Lynx. It's totally usable on something like floodgap.org or oldskool.org. You can view pictures offline, and I recommend the DOS viewer CSHOW because of its great speed and dithering routines. Pictures look okay even in 16 color dithering, and should be very good if you can do 640x480@256 on your card. There's stuff like WInjpeg... but slow. Modern pages use HTTPS, but there's a project here to translate HTTPS as HTTP. Keep in mind you cannot go wherever you want on the Internet. The newest, biggest browser is IE 5.0, but it's no more capable of handling modern pages. Load a page that's not vintage or coded to be compatible with vintage browsers like Lynx and Netscape, and you'll get gibberish on the screen, thrash the CPU and page file, and maybe lock the machine up. Can't even click links without checking the URL to make sure they're safe. But Google works, Wikipedia works, sites coded to work do work pretty well. You'll just need to make bookmarks of what works. Remember those?
Also, gopher is your friend. And Wsgopher is the best client. It works well on any 386 and is kinda like the Web. There's hundreds of gopher holes out there, can even get the weather and daily news via it. Best part is you can go anywhere, click any link with no worry. It's a trove of weird stuff, stashes of personal files, diaries, lives and hobbies put online kind of like a textual geocities. Very much like seeing the Web in 1994 when nothing was commercial. I think Gopher is actually the best case for having the Internet on a 386, and it died out as a mainstream protocol in the heart of the Windows 3.1 era so you'll miss nothing by using vintage hardware.
FTP is the easiest way to transfer files via a network, and CuteFTP is the best client. WSFTP tends to abort in the middle of transfers. MIRC 5.91s will get you on IRC full speed. For Telnet, like to BBSes and SDF, try QVT/net. Teraterm is good but won't give you the proper ANSI retro feel. Telix for Windows is commercial, but if you want the real BBS feel maybe try and track it down.
Honestly, Windows 3.11 is surprisingly usable on the Internet. It can quickly load retro-styled Web pages coded in old HTML, and there's hundreds of user pages out on Gopher. There's all the old BBSes, and some like Level 29 have active boards. With a UNIX shell service like SDF, you can connect to SSH servers and even get email with Alpine. Using this is much more like exploring, like the really early Web was. I highly recommend you try this. If you go Windows 3.11, you get access to professional, good and easy to use TCP/IP software -- and you can multitask it, view a picture while leaving the browser open while running a telnet session, say. There's even a TCP/IP Telix for Windows you could try and track down.
I've written a lot, I know, but I feel like Windows 3.x gets unfairly maligned. It's a pretty capable multasking OS, and even has some good native games like Castle of the Winds, Exile, Civilization and The Incredible Machine. Also, despite its reputation, it's reasonably stable if you have all the updates applied. I think it earned its reputation because people back in the day were running 3.1 with no updates, Trumpet winsock, not enough files or breakpoints, random real mode DOS drivers, too many TSRs and desk accessories. Also, gotta back up the Windows folder regularly. DLL stomping is as bad in 3.1 as it is in 95 -- as many programs use bespoke, poorly made installers. Fortunately, the fix is just copying the Windows directory back -- it's all self-contained.