How can auxin cause shoots to grow up and roots to grow down?

Auxin enables the leaves to turn toward light thereby increasing photosynthesis. Auxin has nothing to do with the roots growing down, they are simply positively hydrotrophic.

Auxins are often used to promote initiation of adventitious roots and are the active ingredient of the commercial preparations used in horticulture to root stem cuttings. They can also be used to promote uniform flowering, to promote fruit set, and to prevent premature fruit drop.

Used in high doses, auxin stimulates the production of ethylene. Excess ethylene can inhibit elongation growth, cause leaves to fall (leaf abscission), and even kill the plant. Some synthetic auxins such as 2,4-D and 2,4,5-trichlorophenoxyacetic acid (2,4,5-T) have been used as herbicides. Broad-leaf plants (dicots) such as dandelions are much more susceptible to auxins than narrow-leaf plants (monocots) like grass and cereal crops.
Auxins directly stimulate or inhibit the expression of specific genes. Auxin induces transcription by targeting for degradation members of the Aux/IAA family of transcriptional repressor proteins, The degradation of the Aux/IAAs leads to the derepression of Auxin Respose Factors ARF-mediated transcription. Aux/IAAs are targeted for degradation by ubiquitination, catalysed by an SCF-type ubiquitin-protein ligase.

In 2005, it was demonstrated that the F-box protein TIR1, which is part of the ubiquitin ligase complex SCFTIR1, is an auxin receptor. Upon auxin binding TIR1 recruits specific transcriptional repressors (the Aux/IAA repressors) for ubiquitination by the SCF complex. This marking process leads to the degradation of the repressors by the proteasome, alleviating repression and leading to specific gene expression in response to auxins (reviewed in [1]).

Another protein called ABP1 (Auxin Binding Protein 1) is a putative receptor, but its role is unclear. Electrophysiological experiments with protoplasts and anti-ABP1 antibodies suggest that ABP1 may have a function at the plasma membrane.
On the cellular level, auxin is essential for cell growth, affecting both cell division and cellular expansion. Depending on the specific tissue, auxin may promote axial elongation (as in shoots), lateral expansion (as in root swelling), or isodiametric expansion (as in fruit growth). In some cases (coleoptile growth) auxin-promoted cellular expansion occurs in the absence of cell division. In other cases, auxin-promoted cell division and cell expansion may be closely sequenced within the same tissue (root initiation, fruit growth). In a living plant it appears that auxins and other plant hormones nearly always interact to determine patterns of plant development.

According to the acid growth hypothesis for auxin action, auxins may directly stimulate the early phases of cell elongation by causing responsive cells to actively transport hydrogen ions out of the cell, thus lowering the pH around cells. This acidification of the cell wall region activates wall-loosening proteins known as expansins, which allow slippage of cellulose microfibrils in the cell wall, making the cell wall less rigid. When the cell wall is loosened by the action of auxins, this now-less-rigid wall is expanded by cell turgor pressure, which presses against the cell wall.[citation needed]

However, the acid growth hypothesis does not by itself account for the increased synthesis and transport of cell wall precursors and secretory activity in the Golgi system that accompany and sustain auxin-promoted cell expansion.Growth and division of plant cells together result in growth of tissue, and specific tissue growth contributes to the development of plant organs. Growth of cells contributes to the plant's size, but uneven localized growth produces bending, turning and directionalization of organs- for example, stems turning toward light sources (phototropism), roots growing in response to gravity (gravitropism), and other tropisms.

The short answer to your question is that the root and shoot systems have tissue-  and cell-specific responses to auxin.  This is due in part to the fact that cells within these tissues express different sets of genes, whose products respond to auxin in different ways.

In shoots, the accumulation of auxin causes cells to elongate. Whereas in the root, the local accumulation of auxin leads to the restriction of cell elongation.  While the acid growth hypothesis may help to explain the effect of auxin on shoot growth, the effect of auxin on the downward growth of roots most surely stems from the activation of genes that function to inhibit cell elongation.

The long answer can be found only by understanding auxin signaling and auxin homeostasis as it effects plant growth.  Start by reading the info on the molecular mechanisms of auxin action in the answer above.