How to produce dry pure oxygen I'm supposed to produce dry and pure oxygen using the decomposition of hydrogen peroxide with a manganese dioxide catalyst.

$$\ce{2H2O2 (aq) → 2H2O (l) + O2 (g)}$$ 

The gas will be collected in a bottle by downward displacement of water. How can I safely move the oxygen gas through the drying agent without mixing in atmospheric gas?
Commercial drying agents, such as Drierite (anhydrous calcium sulfate), or other desiccant, should do the job for small quantities of oxygen.

Most inorganic desiccants should not react with diatomic oxygen; as long as the oxygen is produced separately from the desiccant, there should be no issue. (Dropping 90% or stronger $\ce{H2O2}$ on many substances can cause dangerously rapid decomposition, but with 30% or lower concentration, that danger is minimal.)

As far as preventing atmospheric gases from entering, first bleed the apparatus of residual air by discarding the initial output. Seal the later containers underwater, e.g. by sliding a glass plate across the mouth. See Janice Crowleys' demo on YouTube.