I wouldn't bother setting up a dual boot MS-DOS 6.22/Windows 3.11 & Windows 9x environment.
Generally speaking (without going into the technical details), Windows 95 & Windows 98 still runs on top of MS-DOS.
You can therefore bypass the graphic user interface (GUI) and boot straight into MS-DOS 7.10 (or MS-DOS 7.00 if you're using the earlier versions of Windows 95).
This is a "real mode" MS-DOS (not a DOS command prompt that you launch inside Windows).
Also, unless you have a very fast 486 CPU (Intel DX4-100 or higher), I would recommend that you rather install Windows 95 as it runs a bit better on a 486 system than Windows 98 (Microsoft also recommends a Pentium processor for Windows 98).
In order to achieve the above, you need to edit the MS-DOS.SYS file in your root directory (it's hidden) to bypass your GUI.
You will then also have to edit your Autoexec.bat & Config.sys startup files, which is where you would set up your boot up menu(s) - you can have multiple boot configurations.
I've attached a little guide that should help you on your way (please note: I'm not the author of this guide - just something I grabbed off the internet years ago. So, I can't take credit for this).
There are TCP/IP stack drivers for Windows 3.11, which you can search on the internet but, as stated above, perhaps first try the real mode MS-DOS setup under Windows 9x?