Which Microcontroller to use
To answer your first question, regarding stacking a WiFi Arduino Shield on top of a LCD Shield:


  can I stack them simultaneously


The obvious answer is - it depends on the shields. Each shield might use a different set of Arduino I/O pins, making them unavailable to the next shield stacked on top of it.

for example, this formal Arduino WiFi shield makes I/O pins 10, 11, 12, 13 and 7 (on Arduino Uno) unavailable to the next top shield. 

Unfortunately, this LCD KeyPad shield needs pins 7 and 10 available (among others) in order to function properly. This makes it incompatible for stacking on top of the WiFi shield mentioned above (or vice versa).

It doesn't mean that there is no way that any LCD shield could be stacked on top of a WiFi shield, but to be honest, I couldn't easily find any LCD shield that won't collide with a WiFI shield pins. You might want to change approach and send the WiFi data using the built-in Serial to a computer.


  Is that the best microcontroller to use


If there would have been a manifest to this website, I'm sure that "The best product is the one that meet your needs best" would be it's first sentence. In order for someone with experience to answer what is best, (s)he would need to know your exact needs. Since you are still not sure what they are, and just want to play around a bit, I imagine that the classic Arduino Uno R3 could help you find your way, especially if you are doing your first steps with Arduino.