Duane,
The trail that most tourists want to hike is the Sentiero Azzurro which has four segments. Some of these have been closed for an extended time to due storm / slide damage, which can occur at any time. As Kathy mentioned, only the segment from Monterosso to Vernazza is open at the moment, but that could change at any time and other segments could open. Note that to hike any of the S.A. trails, a Park Pass is required. These can be easily purchased at local stations or even in the small wooden huts at the beginning of each trail segment.
The segment from Monterosso to Vernazza is the most challenging, as there's quite a climb at the Monterosso end. It's not difficult but if you're not in good shape it will be some work. I'm 60+ and can manage, but I have to take my time. Be sure to take water! Once you're up at the level of the trail, it's mostly easy walking, although there are some narrow sections. When you arrive at the Vernazza end, there's a viewpoint overlooking the town, so be sure to stop and get the usual photo.
Although the other segments are currently closed, these may re-open at any time, as the usual "official" start of tourist season is April 1st. The easiest segment is from Manarola to Riomaggiore on the famous Via dell' Amore, which is mostly flat and paved. There's a bar just outside Manarola where you can stop and get a "cool one" and enjoy the gorgeous ocean views.
One other point to note is that there's an extensive network of trails in that area, so it's possible to do lots of hiking outside of the Sentiero Azzurro trails. Many of the trails go further up into the hills, so they're a more robust hiking experience.