Tam and Lee,
As Laurel mentioned, the "official" start of the tourist season in the Cinque Terre is usually 1 April, so the trails may not be open yet. Several of them were closed all of last year, such as the famous Via dell' Amore from Riomaggiore to Manarola.
Corniglia is not the best choice for places to stay, especially in March. As mentioned it's hard to access and also very small. I'd suggest looking for a place in Monterosso as it's the largest of the five towns and therefore has the greatest number of hotels, restaurants and other tourist amenities (not all of them may be open in March). I'm a bit out of touch with "romantic" but if it's open and your budget will allow, you might try La Torretta in Manarola. There are lots of restaurants in Monterosso, and if you were staying in Manarola it's an easy matter to take the train there for dinner as the trip is only a few minutes.
One other issue at that time of year is that the weather could be cool and rainy. If you the weather is nice, you'll definitely see "great natural beauty". These are very small towns, and I can just about guarantee that there won't be any ballroom dancing in any of the towns.