What kind of relay should be used to control a thermostat terminal?
Just to let you know that most all thermostats have 24 VAC supplied by the air handler, which maybe in the garage or an isolated room. What you need is 2 small relays under software control. One to turn on the heat, the other to turn on the air conditioner. Possibly a 3rd relay to run just the blower fan, which some central heat/AC units offer. Only one relay at a time can be ON.

You need a Rpi relay board with 2 or 3 relays to isolate the Rpi 5 volt relay power from the 24 VAC that control the air handler MPU. Google "Raspberry pi relay IO" to get this small board which mounts on the 40 pin header. It is low cost.

It is made for recent versions of the Rpi so all you need to do is write the software in Python. As far as thermostat connections go the wire marked "G" is earth ground. The wire marked "C" is the common return for the control lines which are red and blue.

You can find out as suggested by @Marko Buršič to scan Honeywells website as to which wire does what, or you can connect a short insulated wire from red to common to see what it activates (either heating or cooling). Then use the same wire to try the blue wire connection. Chances are it will turn on either heating or cooling.

24 VAC is not a shock hazard, but I am assuming it is supplied from a small transformer that powers the heat/AC control board. Do NOT have the blue and red wires connected to common at the same time unless Honeywell states that is how you activate just the blower fan with no heat or AC. Your software must be well written so illogical relay states are NOT possible.

One fail-proof option is to use 1 relay to enable the connection to common, so it acts as a main ON/OFF relay. Relay 2 then is used to select either heat or AC. It cannot select both at the same time. If Honeywell documentation states that is how a "fan only" mode is achieved, then use a third relay to do this.