This 6.8 Mile (11 Km) hike traverses through the Lavaux vineyards on gravel and concrete pathways.  It begins at the Lutry train station and winds through vineyards and small vineyard making towns until it ends at the Saint Saphorin train station.
The walk is a lot of up and down throughout with the first mile being almost entirely uphill and the last mile mostly downhill.  The middle section has short bouts of up and down with small straight areas mixed in as well.
I have a group of friends who also have young children at home with them.  We decided to start a "Tuesday Tours" group which focuses on walks and exploring new areas with toddlers in strollers of backpacks.  This hike was done pushing two toddlers in strollers the entire way.  There were two sections that required a bit of effort with the strollers.  The first was a set of 3 stairs which required us to take turns carrying the strollers down the 3 steps.  And the second was a narrow area where the rear wheels of our strollers did not fit on the path:
Fortunately this narrow area was not very far (maybe 15 feet).  And if you had a smaller stroller (umbrella stroller) I think the wheels would have fit.  The walk follows the lake and offers great views of Lake Geneva with the alps jutting up on the other side.
We did a smaller portion of this hike on a clear Spring day as a family.  The backdrop of the Alps on a clear day is magnificent:
The walk is well marked with signs titled Terrasses de Lavaux and yellow arrows or walking man stickers in between:
If possible, I would recommend this hike in late Summer to early Fall as the grapevines are full during this time of the year.  When we walked this route in the Spring it was also beautiful, however when it is close to harvest time the vineyards are so beautiful.
 Along the way, we passed old stone homes centered in the vineyard they owned, castles off in the distant hills, buckets and buckets of grapes (almost as tall as me) that had just been harvested and even a helicopter that made multiple trips to pick up large buckets of grapes to transport them up the hill.
About 2 miles from Saint Saphorin there is a little hut that serves wine and grape juice from the local vineyard.  A well-deserved stop after a good portion of the walk and nice to have taste of what you have just walked through:
The views of the old villages set between the vineyards are extremely charming and well-preserved:


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There is also an option to do part of this trail on a small train on wheels.  If walking this far isn't doable or if you have multiple children, consider the Lavaux Express Wine Train.  My children really enjoyed it!