Do you want to filter traffic between the devices? Probably not possible with the stated goal.
Do you want different devices to be on different subnets? Reads as no.
Do you want all the devices to be on the same subnet?
Routing works by having different subnets on each side of the router. You can use a router as a switch in many situations by not using the WAN port at all ... just disable all services on the Linksys, then plug in the different devices to the LAN ports, then connect 1 of the LAN ports to the Verizon router. This will end up with a flat network. With all the services on the linksys disabled, the Verizon will need to provide those. If DHCP is disabled on the linksys, then addresses need to be either manually configured or DHCP provided by the Verizon.
You'll want to understand networking better to do much else. As long as you are changing things, it really is best to never use 192.168.1.1/24 as a network anywhere. Same for 192.168.0.1/24, 10.0.x.x/24 and 172.16.x.x/24. Avoid those so you can have flexibility in the future for remote connections from a cafe or hotel. The selected subnets still need to be in the RFC-approved private network spaces, but there are thousands of those.192.168.51.1/24 wouldn't be a bad choice, but really the "51" can be anything from 3 -to- 254 to achieve the goal. The goal is never (as close to never as possible) be on a remote network with the same subnet.
To learn how networking works, search for episodes 25 of the
Security Now podcast. It is the big picture ideas you need to understand. Details can be looked up as needed.
Have you considered using just a wifi AP instead? Ubiquiti makes some for $50-$300 with PoE that can be placed almost anywhere. Because the power is over the ethernet cable too, we aren't restricted to locations with an electrical outlet. Consumer routers tend to lose vendor patch support in 3-4 years, so they really shouldn't be on any network. But Pro-sumer devices like MikroTek and Ubiquiti sell are maintained much longer. Plus, wifi-standards are constantly changing so to get better, more secure, faster connections, we need to change wifi more often. A bunch of new wifi gear is about to be released to support the newer standards, Wifi-6 (802.11ax) for example.
https://www.howtogeek.com/368332/wi-...hy-it-matters/
If you need to get networking to different parts of the house between two APs/routers, I can suggest powerline networking, just expect about 10% of the bandwidth that the box claims. So, my 600 Mbps powerline going between the home-office and downstairs to the opposite side of the house really gets only about 60 Mbps, but at least it is stable and encrypted. On the same floor, between a room and hallway outside (shared wall), the bandwidth was 220 Mbps ... so for my house, the box's claims are pretty aggressive. They make 1200 Mbps powerline pretty cheap these days. It is stable bandwidth, not like wifi, which fluctuates massively. We seldom need wifi where our routers are located. Think that's a universal problem.
I'm a little more security conscious than most, so I keep wifi off my main network. Wifi can't really be trusted.
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