The single best day trip from La Serena is up the Elqui Valley — pisco distilleries, the home town of Gabriela Mistral, and the clearest night skies on the planet. Allow a full day, or two if you want to stay over and see an observatory.
Other options include the marine reserve of Pingüino de Humboldt up the coast (full day, leave early), and the seaside resort of Coquimbo just across the bay — though many travellers find it less interesting than its larger neighbour. Hidden valleys further north — Hurtado, Limarí — repay anyone with their own transport and a couple of extra days.