I'm traveling there from 03/26-28 for business and would like to see a few things, one of which is the St. Louis Botanical Garden.
They have an official "What's in Bloom?" page!
For March, you may be lucky enough to see cherry blossom, as well as crocuses, vines, daisies and more.
Of course, like with any plant life, if it's unseasonably warm or cold the bloom may happen slightly earlier or later, so check February and April as well for an idea of what might also be occuring.