Andros is well-connected by ferry and sits close enough to several other islands and the mainland for easy day trips — though the island itself rewards at least three or four days, and most visitors find they have barely scratched it.
The obvious destination is Tinos, 30 minutes away by fast ferry, with its pilgrimage church of Panagia Evangelistria and a remarkable village culture in the interior. Tinos is the antithesis of Andros in atmosphere — deeply religious, busy in summer with Greek pilgrims — but the two islands make a natural pair.
Syros, the administrative capital of the Cyclades, is about an hour away and worth half a day for its extraordinary neoclassical architecture, opera house, and the contrast between the Catholic hilltop town of Ano Syros and the Orthodox port below. The ferry connections are frequent in summer.
For those wanting to reach the mainland, Rafina is the port for Athens: a fast ferry takes around two hours, which puts the city within reach. Rafina itself is an ordinary working port but has good fish tavernas along the waterfront if you need to kill time before the return crossing.