When we talk subnet, we are talking IP addresses. IP addresses being the address assigned to each machine on the network, so they know which information is for them when the information is sent across the network.
An IPv4 IP address, which is what we mostly all use here in the US. It looks like this, ###.###.###.###
The number in each section can be anywhere between 0 and 255, the first set has to start with at least a 1.
Everything is in the same subnet if they have the same numbers in the first 3 sections, and they will be able to "see" themselves on the network. If there is a different number in any of the first 3 numbers, they ignore the information considering it separate from their own network.
There are a few ranges of IP addresses that are reserved for private use, the rest are used on the internet for addressing. Typically, most people use 192.168.X.X, specifically the subnet 192.168.1.X or 192.168.0.X
A router will offer DHCP service to your network typically, what this means is if the device is setup for dynamically assigned IP addresses, it will ask the DHCP service on the router for its own settings to use on the network. typically on home devices this will be between 150-200 on the same subnet as the routers internal IP(because it will have an external facing one for the internet to see it, because internet is different from intranet(your private network).
So that is a range of 50 devices that will work already setup, you can change the range in the routers web interface, by opening a browser on a device on the network and putting in the IP address of the router(typically 192.168.X.1)
Now, the router is the core of a home network, everything typically connects to it, and it connects to the modem which is your portal to the internet. most home routers only come with 4 LAN connections, the WAN connection being reserved to connect to the modem/internet.
If you want to connect more devices, typically you'd connect them wirelessly. But you can also use switches, or configure routers to connect. Configuring the routers incorrectly and the routers will either cause IP conflicts or assign devices to another subnet and not allow them to connect(or just cause an issue with NAT(which is devices accessing things behind a router). If you are using a dumb switch(which is easiest), 1 cable between switch and router LAN port, and everything else can connect to the switch, and done. If using another router, you want to disable DHCP on the router, set the internal IP of the router to Dynamically get an address, or an IP address in the same subnet that isn't in the range DHCP serves from the main router. Then hook everything to the LAN ports(including the main router), ignore the WAN port. This will keep everything on the same subnet and communicating properly.
That is a basic overview and may help you understand what you need to do.