  Using a variational approach we study regularity properties for
quasiminimizers of a $(p,q)$-Dirichlet integral inside a bounded open set, as
well as regularity results up to the boundary, in the setting of a metric space
equipped with a doubling measure and supporting a Poincar\'e inequality. For
the interior regularity, we use De Giorgi type conditions to show that
quasiminimizers are locally H\"older continuous and they satisfy Harnack
inequality, the strong maximum principle, and Liouville's Theorem. Furthermore,
we give a pointwise estimate near a boundary point, also a sufficient condition
for H\"older continuity, and a Wiener type regularity condition for continuity
of quasiminimizers at a boundary point. Finally, we consider $(p,q)$-minimizers
and we give an estimate for their oscillation at boundary points.
