I'm new to this, so maybe I'm missing something.
I've seen quite a few LCDs that plug directly on top of an Arduino Mega (or other Arduino boards - I'm interested in the Mega), but they all seem to cover all the I/O pins and do not extend them.
Here's an example (I know that this is not a genuine Arduino, but the photos show what I mean):
Mega 2560 R3 Development Board + 3.5 inch TFT LCD Touch Screen Display Module Compatible For Arduino Mega2560 R3
How can I access the pins once the LCD is installed?
Your best bet is to get an lcd and shield from an outfit who know what they are doing. Try
It pretty hard to get right for everybody but there are several options. My 240x400 shield uses all the pins at the end but only pins D2>D7 at the side,and they provided bent headers to get at other stuff. That shield carries an SD card, which is commonplace.
You can assemble the sensors etc. on a proto shield and have that immediately above the Mega.