- adequate bypass caps on the microcontroller.- reverse diodes across the relay coils.- snubbers on the relay contacts [do google search on "snubber"].Also, in general, you cannot drive a 5V relay directly from an Arduino I/O pin, as 5V relaystypically require somewhat more current than the pins can provide, so people typically use NPN transistor inverters between the I/O pin and relay.
I'd also add more supply decoupling on the 5V rail right next to the relay and its transistor. 220uF or so. Replace that 1N4007 with a fast recovery rectifier, its too slow.
You mentioned that you have no bypass caps. That would suggest that you think it would really be practical to use them but you either didn't bother, or couldn't be bothered. Add a larger cap to the power supply side and some smaller ones to the display and the lines going to the relay control.
Replace that 1N4007 with a fast recovery rectifier, its too slow.