Two day trips are the obvious ones from Asmara: down the spectacular descent to the Red Sea port of Massawa, and west across the highland plateau to the market town of Keren. Either makes for a long but rewarding day out, though Massawa really wants an overnight stay because of the heat and the drive time.
Closer day excursions include the small monastery at Debre Bizen above Nefasit (around an hour east, with a steep climb on foot) and the village of Filfil-Selomuna at the edge of the green belt that descends toward the coast. Both need a sturdy vehicle and a travel permit.