How to connect a 240x320 37-pin TFT to an Arduino Nano?
The Nano isn't a good choice for driving that screen. You need almost all the pins of the Nano to connect to the screen alone, and there's not enough to include the touch-screen as well. And that's when you're not using the serial port of the Arduino (TX/RX pins) for communication with the PC.

You need to connect:


16 data pins
/CS pin
/RS pin
/WR pin
Optionally RD pin, or tie it to 3.3V
/RESET pin


And all those need to be level shifted to 3.3V.

You need to provide 3.3V to VCC1/2 and VCI.  The LED A connects to 5V (since it has more current capacity than the 3.3V) and each LED Kx pin through a suitable resistor (220Ω maybe) to ground.

The touchscreen needs the X- and Y- pins connecting to a digital IO pin, and the X+ and Y+ pins to analog IO pins that can also double as digital IO pins (so not A6/A7 on boards that have them). 

And then you need to obtain or write the software to drive the ILI9320 through the 16-bit interface.

Driving the interface is simple enough:

Initially:


Set /CS HIGH
Set /WR HIGH
Set /RD HIGH (if not tied to 3.3V)
Set /RESET HIGH
Short delay
Set /RESET LOW
Short delay
Set /RESET HIGH


Then to transfer a group of 16-bit words:


Set /CS LOW
Set the value to write on the 16 data pins
Pulse /WR LOW then HIGH 
Repeat from 2 for each word
Set /CS HIGH


Refer to the ILI9320 datasheet for how to control the chip, or use existing library code (does uTFT or u8glib support it?).

For the touch screen it's a case of setting Y- LOW and Y+ HIGH, and X- and X+ to INPUT, then reading the analog value from X+.  Repeat with X- set LOW and X+ set HIGH, and both Y set to INPUT, and read the analog value from Y+.

You can also get better results by reading multiple inverted settings:


Y- = LOW, Y+ = HIGH, X- = INPUT, Read X+.
Y- = HIGH, Y- = LOW, X- = INPUT, Read X+.
Subtract the second reading from 1023
Add the first and (modified) second reading
Divide by two.
Repeat for X-/X+ reading from Y+.


You can also take multiple readings and average them out to reduce noise and stabilise the output somewhat.

And of course on top of all that you need to find a way to connect to a 0.5mm pitch connector. Normally a 0.5mm pitch FPC socket is used, which means a PCB with one on.  You may be able to find 37 pin adaptor for it, but 37 pins isn't a common format.  Normally 50 pin is more common to accommodate a more general purpose interface including the M0-M2 pins that allow you to select the interface type, as well as an 18-bit data bus to allow full RGB6/6/6 interfacing, and even buffer-less clocking of raw display data for very high speed display updates by MCUs that support such an interface.