The two big day trips out of Meru are Meru National Park to the east and the trailhead of the Chogoria route up Mount Kenya to the south-west. The park is a long day from town — an early start, several hours on dusty roads each way — and is better visited on an overnight trip if you can manage it. Chogoria itself is just under an hour by matatu and worth a stop for the views west into the Mount Kenya foothills, even if you are not climbing.
Closer to town, the village of Kibirichia on the upper slopes has tea-farm tours and trout fishing in the streams that run down from the bamboo zone. Easy half-day trip; arrange transport in Meru.