What Microprocessor with Wifi or Radio Frequency To Use

Ok. An esp8266 averages around 80mA. So your battery needs to be at least 640mAh. You could use 3xAA cells or 4xAA rechargeable cells. They have a capacity of around 2000mAh. Or a 18650 size Li-ion battery which will have a 2000-3000mAh capacity. I recommend a Wemos mini. The standard or Lite version should have enough range, but if not, you can get the Pro version and an external antenna with a higher gain. 2x waterproof ds18b20 temp sensors. Simples!