Microcontroller requirements for TFT display
Since the display includes the Ilitek ILI9320 controller, then your interface requirements are much lower, as the microcontroller no longer has to interface directly with the TFT and instead only talks to the controller chip via a simple interface: either SPI, which takes six wires: RS, CS, CLK, MOSI, MISO and RESET.  Or you can use an 8080-compatible parallel interface which takes 13 wires: an 8-bit data bus, and RS, CS, WR, RD and RESET. (There are options to use larger data-buses, up to 18 bits, but I don't recommend that for a low end microcontroller.)

There are two optional interfaces in which the microcontroller generates all of the clock signals (VSYNC, HSYNC and DOTCLK); you don't want to do that since it would require a high-end controller.

So just about any microcontroller will do, however you need to have enough flash memory to hold whatever static items you want to display; for example if you are going to be displaying text then you will need to allocate arrays to store bitmaps for whatever fonts you will use.  Even a small font can take 60KB.

One of the advantages of this controller is that includes 172,800 bytes of RAM, which is enough to store 320*240*18 bits.  However it is not double-buffered, meaning that when you write to the RAM, it will immediately show up on the screen, and if you updating a lot of the screen, it will be noticeable.