Technically, what you wrote above is the dress code which can be enforced at any time. However, most of us wear jeans, plain t-shirts (either with a Mizzou logo, no logo, or a dressier shirt of some kind... you're probably fine as long as it doesn't have a logo for another school on it or holes in it!), and flip flops/sneakers. If at any point we happen to piss off the heads of the school, however, they can certainly see to it that the real dress code is enforced. My class has been warned several times about it now and they're threatening to pull it on us due to bared shoulders (a big no-no) and jeans with holes in them (from wear and tear or directly off the shelf at the store), so do be mindful of that. Otherwise, it's very laid back.
As for scrubs, you really only need one pair that you're willing to throw out after your first year, and you might want to get a lower quality lab coat for the days when you can't bear to put on the nastiness that those scrubs will become. Most people just take their stuff home to wash once a week, and any more than that will be a waste of time, but everything you own will smell. I feel your pain in advance! Really don't spend money on nice scrubs for this. We have school scrubs here that everyone fell in love with and bought for lab this year, but now they're ruined and have to be replaced, so that was kind of wasteful lol.