Measuring air temperature requires:
1) Decide on budget/cost.
2) Decide on accuracy required. Balance budget vs. accuracy. For best accuracy you will need to ventillate your sensor when making air temperature measurements.
3) Select an appropriate sensor, data acquisition, and display.
4) Draw schematic. Prototype sections that you are unsure of.
5) Layout PCB. There are free software packages available, just look...
6) Assemble and troubleshoot if there are PCB layout mistakes.
For a simple, yet accurate temperature measurement you could use a thermistor like the YSI44031 (0.1 deg C interchangable) in a half-bridge configuration. I would select a micro that has an integrated ADC to do the data acquisition. There are many options for the display, kinda depends on what you want the thing to look like, and your budget.
Another option is to select a solid-state temperature sensor like the LM35 which already has a linearized output, but would not be as accurate as the YSI thermistor. Analog Devices makes some solid-state temperature sensors that are very accurate, but they cost a lot. You may be able to acquire samples though.
So many options... Start with #1 and #2 above.