Cinque Terre

You need to check the National Park of The Cinque Terre website before you go. There has been some damage in Liguria from the rain even this week. The entire region is geologically unstable due to the steep terrain along the coast.
Any town is a good base. Monterosso is bigger and offers more amenities and more trains (also better beach).

Vernazza was worst hit by the storm in 2011, but we were there in May 2012 and the reconstruction was amazing and it was in pretty good shape. At this website you can see that essentially everything is now open again: http://savevernazza.com/visiting-vernazza/. If you want a resort town with a beach, stay in Monterosso (but there is also car traffic there). If you want a charming little town next to the water, stay in Vernazza or Manarola. If you want more peace and tranquility, up on the cliff, stay in Corniglia. Riomaggiore is also charming and next to the water, but somehow seems less charming to me as compared to Vernazza or Manarola. Vernazza is very popular and apparently can be quite crowded in the summer months -- October should be very nice. We have stayed in Vernazza and Manarola, and both were very nice. Regarding trails, I think I read somewhere that all the trails are open now except for Riomaggiore to Manarola, but this website says otherwise: http://www.parconazionale5terre.it/. But don't worry, there are multiple trail options and they are all good. Btw, I just booked some rooms in Vernazza for September, and my preferred options were already booked. I suggest that you do your research and book your room(s) ASAP.

We like Manarola as a base. Beautiful sea views, not as over-run as Vernazza, but all the villages are but a few minutes away by the local train service. We stay with Eraldo & Donatella at "Da Paulin" apartments. Wifi & great views. RS certified, and a wonderful host couple. We have stayed 3 times with them, last 2 times in late September. Was sunny & warm -- and uncrowded -- for both of our visits.