Which cities should we try to visit for our day trip?
Depends.... You can easily visit 3 if not all 5 of the Cinque Terre. Rick Steves has self-guided walks in each of the towns that we thought were great. Each takes 30-60 minutes, so if you just want to get a feel for them and see some great views, you could take the train to Manarola and visit each town, working your way north. Be aware it takes about 1.5 hours to get to Manarola from Genoa. Pick up the R.S. guide if you do not have it. I think there is a "pocket guide" for the Cinque Terre.