Undoubtedly a poor experience - but thats half the fun. Im a creative in the design business - I have fond memories of some of the old software. A bit sad I know.LOL!!
So... question... what is your ultimate goal here? Browsing the web will be.... a poor experience on the Classic. But maybe you want to use it as an email client, or do direct downloads of small files or whatever?
The second thing is, do you have the Daynastar adapter software? Here's a conversation that discusses it, if you need it. you would need a way to get it onto your Classic: https://68kmla.org/forums/index.php?/topic/17016-drivers-for-daynascsi-adapter/
The next thing... can you get the networking information you need from your router? You will need the information because you will need to enter it manually into a control panel. You will need the Gateway (router IP), an IP address, the subnet, and the DNS information. If you have your OSX Mac handy, you can check Network Settings and copy them so as to type them into the Classic later (you will need to choose a different IP address though. Your router is probably using DHCP. Just pick a number somewhere far down in the range.
I had no luck with the default MacTCP and 7.5.5 trying to connect to the web. It was fine for AppleTalk, but I couldn't see the world until I installed Open Transport. It's just an alternative to MacTCP. Again, you would need a way to get it onto your Mac.
So.. make sure you have the right software and the right settings. If you can't get the software onto the Classic, you should get some help from someone who has an Intermediary machine, or get your own. I would recommend a PowerPC machine... one that has a built in floppy drive because floppy drives are useful when trying to get software (like Daynastar software) onto old Macs.
My Classic has had a large capacity drive fitted - replacing the 20Mb, its got Gbs. It has 4Mb of memory. I had all the Daynastar software on floppy. Im cannot remember if there is open transport or Mac TCP. I will check.
The confusing bit for me is this - I got a Netgear switch, assuming that as it handled 10BaseT as well as 100 and 1000 that would carry the signal fine. Now I have my doubts.
The there is the issue of networking and having access to the internet - two entirely different things. Im not sure if the switch might give me access to the internet but not my other machines.
Im considering getting an iMac G3, which has like OS9 or something. This might enable networking at least.