Graphical LCDs gives a huge advantage in embedded project. When it comes with touch screen things get more interesting. Zizzle shares his tutorial project on how to set up STM32 microcontroller to work with 320×240 LCD with touch screen. As he states running such LCDs on smaller scale microcontrollers like AVR (Arduino) is possible but eats to much of resources without leaving them for real applications.
Using ARM is what at least is needed along with additional memory for buffering screen images. You can choose lots of boards that are already built for you like these on ebay based on STM32F103, Project setup consist of Ubuntu where ARM toolchain is running. Eclipse C++ IDE is obvious selection. Openocd was chosen for programming and debugging. Another nice accent in code that it is FreeRTOS based. So this gives a great template for further development of various projects.