You might consider using these modules, then you can keep the Uno.
Thanks... while it would be cool to have WiFi/online functionality, I think it might be an overkill and I don't really need to control anything from far away. The heating system gets turned off anyway when there's nobody at home. I was more interested in getting better temperature precision and better sampling rate/response time. Which I already got. I just need to make the relay and sensor wireless, not necessarily WiFi.
I will look into it.
There have ben a lot of responses but it would help all of us if you would post links to the actual devices you intend to use.
Thats what happens if we arent certain what you actually intend to use.
It's been mentioned multiple times. Even with photos. Maybe scroll up? 
Wasted my time looking for a link. Wont do that again. FYI a photo isnt a link.
K, here's the link:
Amazingly google does help. The first link I found
thejn
First, but low quality... Like often found in Instructables.
Who uses Esp8266 AT-firmware at these days?
The relay module tutorial has probably damaged many Esp-boards not mentioning beginners one word about usb-ttl converter voltage compatibility/selection.
Perhaps the OP could look for more links themselves?
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