You plug one switch into the other switch using a regular network cable, and that one into the router again using a regular network cable. Your switch may have a special colored port (usually port #1), if it does that's the one you need to plug the other switch into.
You don't typically need to worry about speed as switches today are smart and if you're communicating between two computers in the same switch it won't go to your router or other switch. Also, switches communicate between themselves and the network at 1000 Mbps even the cheapest ones today (if it doesn't you have some ancient ones)... your internet speed is only 5-25 Mbps typically so your switch is 40x - 200x faster than your internet so the limiting factor will be your internet speed connecting two switches into each other.
Usually you can only connect 1 network wire between switches, if you have an idea of connecting 2+ wires things can happen. The switch may no longer work because you're creating an endless echo/loop (called a loopback condition) until you disconnect one of the lines. Or the switch may detect the loopback and disable one of the wires *(it may use it for backup if the other line fails). Lastly if you know what you're doing and the switch supports LACP (Link Aggregation) it will split users between the lines. LACP isn't perfect, it's more of an "Even numbered birthdays use the left wire and Odd numbered birthdays use the right" which in a crowded environment typically splits it close to 50/50 but in a small environment you can have a bias.
It sounds like just a single wire connecting the switches together will be plenty.