You are welcome.
The S-Bahnen are commuter trains operated out of the main Swiss rail stations or in particular region. S7 simply means that it has an arbitrary number 7. That refers to administration, official time tables, etc. "Wine Train" is a "tourist name" for that train between Puidoux and Vevey. On a map of the whole system, that is also route #111.
The train runs as a shuttle between Vevey and Puidoux. it is on a separate line leaving Vevey, and finishes at Puidoux. If you wish to go on from Puidoux to change to the local train that runs between Lausanne and Palezieux (and beyond, I think), and you can pick up main line trains at those two stations. That local train between Puidoux and Lausanne stops at Grandvaux Station.
I did not find it easy to follow the paths on the Google Map. You are broadly correct, but the devil is in the details, especially when many options involve climbing or descending, only to climb again. Try MapSearch. CH and refer to the scans that I sent. By the way , if you wish, you can buy those maps at many many convenience stores in Switzerland, as well as book stores in our near major train stations. (Ask at Touirist Info.) Even the local stores almost everywhere often have a few maps, because the Swiss really are hikers. They are not cheap- the 1:25,000 topo will be 15-20 CHF and the Wanderkarte even more.
Many hotels lend those maps to their guests, but expect a bill if you do not return them. Or, they may have them at reception to look at.
Mapsearch.ch is much better at showing details for Switzerland.
You can open "Items of Interest " menu and get it to show train stations at various zoom magnifications.
I recall you can get ti to show marked trails.
For immense detail, try Swisstopo:
map.geo.admin.ch/?X=149760.00&Y=546055.0 0&zoom=7&lang=en&topic =swisstopo&bgLayer=ch. swisstopo.pixelkarte-farbe
It appears that you may have missed the best part of the trail just above Lallex, including the part where it passes under the railroad, and I don't know the exact route of the trail along the expressway that you show after Puidoux. it might be better to descend to Epesses as in intermediate point. Then, consider and decide whether to climb back up above Grandvaux station to pick up the trail above Lallex,, and descend again from there. After Lallex, you descend eastward to the the village of Grandvaux, which is quite some elevation below STATION Grandvaux.There are many ways to go through the vineyards, and, unless you follow a Wanderweg, with good trail signs) or pay close attention to some kind of map, you may end up with a lot of unnecessary and tiring ups and downs.
The details available if you zoom in on the Swisstopo link show just how confusing it can be, because the vineyards require many small paths for access. That is a real benefit of following a well marked Wanderweg, such as the one from Chexbres to Epesses.
It may be too much total hike for a 6 year old,or, for you. Are you experienced hikers? The distance is not the main problem, it is altitude changes.
Your idea of locating train stations to shorten the hikes is very appropriate. Unless you are well trained in good condition for walking, including your six year old, you may need to cut the hike short before Villette, for example at Epeses, or at Station Grandvaux., and the part after Villete to Cully is the easiest part.
By the way, the reason there are vineyards on those slopes is that they get a lot of sun in the summer. It can be very hot in the sun, reflecting off the rocky soils.
<<"....Tour de Gourze as I am not sure if my daughter will be able to climb that steep path">>
Most of the route to Tour de Gourze from Puidoux goes by roads, but the last short part is a very steep climb upwards as you noted,and as you can see on the Wanderkarte map scan, comparing it to the fine detail on the Swisstopo map linked in this post.