I'd just submit it to your usual routine for a few days and if there's errors then send the hardware back. If you want to check the stability and it's no inconvenience then sure do it but I don't usually bother as systems are generally fine unless overclocked.
Do it. If anything is wrong you want to find out before you send rebate claims or throw out the packaging or warranties expire. Run memtest86 and/or prime95 and some HDD diagnostics tool. To test the video card just play Crysis or whatever. If your parents say you're playing too much explain you're testing the hardware.
I submit all my builds to some moderate testing (like a test drive), writing a document listening to my mp3s and a video encoding task at the same time would provide a very good test for a stock machine, also running a CPU stress test for 15 mins just to check if the temps are in an acceptable range, no more than that.
Extreme test like 8 hours prime or overnight memtest are for heavily OCed machines.
So in short, yes, I'd do 1 MEMTEST pass, 15 mins Prime95 and half an hour of heavy usage, it should't take much time and it will save you a lot of trouble in the mid/long term.
I would at least run memtest86+ to see if it passes at stock volts, timings, etc. If you don't overclock, don't stress test it. Although, I would try to do normal operations on it like gaming, reading news to test for any obvious problems.