I have just got into Arduino/rPi stuff recently. Def Arduino (actually, ESP32 as it already has built in WiFi!) is the way to go, like you say a Pi would be overkill, plus you’d need an external ADC for the anemometer reading. (PiHut has a ready-to-go anemometer for £42, I did google and found someone from MIT who’d built his own from spoons but by his own admission calibrating it would be a bit tricky 😂)
Don’t see why wind/rain would be tricky? Anemometer as above for wind, you can get a water level sensor on eBay for under £2, just stick that in a pot? (You might want to have some way of automatically emptying said pot I suppose, unless you just reset it manually).
There’s loads of projects on Arduino.cc, I also like the site randomnerdtutorials as they explain things really well. Here’s a basic weather station: https://randomnerdtutorials.com/esp32-web-server-with-bme280-mini-weather-station/
The great thing about the ESP32 is that it can host it’s own web page really easily so you can use your phone to control it.
The “Arduino” FB group is really helpful too, about 130k members so someone clever normally comes along pretty quick to help you out!!