Well this thread made me wax nostalgic. All the other software recommendations are good but for me a must-have when I had Windows 3.1 was HP's Dashboard, a Program Manager replacement.

It was a start menu style launcher before it became a Windows standard, had built-in multiple desktops and resource meter.
While on the topic of setting up Windows 3.1 with TCP/IP there won't be much you will be able to do with it. You won't be able to connect to any websites with HTTPS except Google and Vogons.
Netscape Navigator Gold 3.01:

Internet Explorer 5: (it was miserably slow but rendered slightly better than Netscape)

Opera 3.62 gave me a General Protection Fault when I attempted to connect to Vogons.
The modern internet is far too bloated for Windows 3.1 browsers.