An ESP8266 with temp sensor and a script to pull public weather station data with your local temperature readings into a text file is likely the cheapest way.

What you’re looking for is a 2 channel temperature data logger, but it might be cheaper to just buy 2 single channel ones (a really brief search shows some for like $30)

I don’t think there is any that can record and export data at that price point. I had similar issues in my home and ended up using an Ambient Weather WS-10 display You can get them with various numbers of remote sensors and you a buy additional sensors separately.

I put one sensor outside and scattered the rest throughout my house. That meant I could look at one display and see how the heat or cold was being distributed. It made balancing things a lot easier.

Best of luck!

Just get 2 of these:

https://www.amazon.com/dp/B09DPCW47P/

Can pair with your phone over Bluetooth, or if you’re feeling adventurous, you can spend the rest of your budget on a Raspberry Pi and install Home Assistant on it - it’ll pick these things up automatically, and you can start on your quest of understanding your home with one of the best tools out there.

Snag a 4-pack of AAA Eneloops & a charger, and you won’t have to worry about disposing of batteries for all of your new sensors.

You can also swap one of their outdoor ones with an indoor one with a screen so you can view at a glance - just look for one that takes AAA batteries.

Take a look at the thermometer/hydrometer from SwitchBot (also available on Amazon). $14 each, but available in packs of 2/4 at a discount.

Nearly all SB products seem to have data export from the app. I have three temp sensors (inside/outside/garage), and also use the smart plug mini to monitor and store power consumption data for devices.