IIRC Embedded C and the Atmel AVR includes a weather project. More importantly for you, it introduces the C++ programming language. Since you already have some electronics ability the main difficulty, IMHO, will be climbing the initial near-vertical learning curve of C++.
The ultimate choice of processor might be better made after you know what else will be connected. Will the display be direct driven by I/O pins? IIC? Do you want remote accessibility? And so on.
Don't start out trying to build your weather station. Rather, get your feet wet with basic programming constructs like doing several things at once and switch debouncing. There's also a selection of prewritten examples in the IDE.
If you've got a couple of switches and resistors and LEDs lying around you can do some useful experimentation and there are any number of kits available with a collection of parts for doing all kinds of breadboard projects.
YMMV