I want to be able to control a vacation home's temperature (heating and cooling) via the internet (it has a full-time internet connection).

I've purchased equipment which will allow me to control relays via the internet.

Rather than purchase an expensive networked thermostat, I plan to install a second thermostat next to the first and switch between them using relays. That said, I need to know how to wire everything. Can I simply wire the two 'stats terminals to one another and just use a relay to switch the hot wire coming from the transformer between them?

Yes that will work because the thermostat with the complete line will be the one in control. It does not make a difference which one of the two wires you choose to switch but it has to be the same on for both thermostats.

Power relays by separate circuit wires and separate power supply to prevent problems.

Thank you. I think you answered "yes" above, but just to be crystal clear, if (as I expect) there are more wires connected, it is OK to connect those in series to the 'stats?

You connect the wires just like you would to a single thermostat except you have two of them. It is the relays job to decide which thermostat controls the furnace by the switching action.

As I said before you only need to break one same wire from each thermostat to allow the relay to control. All the relay does is activate the wire going to the appropriate thermostat. This is the most simplistic wiring you can get.

Note. Due to the additional length of wire and the additional relay contacts it is possible you may have to adjust the heat anticipator on both thermostats if so equipped. Do a search here to find out exactly what that entails if you find your thermostat not keeping the correct temperature. If using electronic stats you will probably not need to adjust. Good luck on your project.

Extra note. If you do not know what you are doing I recommend you hire it done. No need to fry something in the process.