Since I work in the IT industry, I am comfortable having what I consider an ideal setup, which is a small virtualization cluster with software firewall/routing combined with switches and wireless access points that support VLANs. So in that case it's easy to isolate equipment in any desired arrangement. For example, all of my cloud-managed devices are isolated and can only access the internet because they are black boxes and untrusted.
However, it's understandable that not everyone is going to be willing or able to put together something similar on their own.
One relatively easy solution is to set up what is called double NAT, where you simply connect a second home router downstream from your main or ISP router. The WAN port of your second router is connected to a LAN port on your first router. Place your old test equipment on the LAN of the second router, and place all your modern home devices on the LAN of the first router. The concept here is that your main home devices are high risk, since you constantly use them to access the Internet and could be hit by some zero day or unpatched attack via web browser or e-mail or etc.. But they are not able to reach the test equipment, protecting those from easy attacks that have been known for decades. (Side note, be sure each router uses a different private IP address range.) You can poke holes in the second firewall as needed to access the test equipment, preferably via some secure method.
Another option that is a step up from there would be to buy a router with multiple interfaces, such as this:
https://shop.netgate.com/products/1100-pfsenseYou can isolate your test equipment on the OPT port and set up specific rules on what can talk to what, similar to the previous example.
Accessing various random websites from an old device running an old browser like IE is definitely a bad idea. Compromised websites and advertisements ARE out there. Many websites run an excessive amount of questionable third party code, such as tracking/analytics/plugins/etc. which could also be an entry point for compromising your test equipment. Modern browsers have a lot of extra protections.