As others have said, the most popular starting point and have it still "count" is Leidy while starting at Chapeta will give you a slightly abbreviated way. To do the
full thing I think you're alluding to, if you're going by USGS maps, then yes, you could start at the junction of FR-47 with Highway 191 north of Vernal. Allegedly, that is where the official trailhead is for the eastern end and then weaves through all the roads as you go westward, but from what I've read elsewhere in the past, because it isn't traveled and maintained as much, it becomes hard to distinguish through some of the "parks" and many of the few people who start with that leg end up going cross country and it can become easy to go off course in a big way if you're not careful. This is where a good GPS or map and compass navigation would be good to have. Then again, perhaps the USFS has done more trail work through the area more recently to improve the route. I don't know. I've also read that it's a lot less scenic and water sources through that stretch between 191 and Leidy may or may not be a little bit more scarce, depending how dry the water year has been. That said, if you're feeling that ambitious, it would be awesome to see a trip report of someone who successfully went that full distance from 191 to 150 or visa versa. The only report I could find online of someone who started from 191 ended up having to pull out at Painter Basin and had to be picked up at Henry Fork due to a bothersome knee or something. I would not attempt a trip like this without some kind of SPOT or InReach device, even if I was with a group.
I think most people will plan for about 6 days if starting from Leidy. Shave a day if starting from Chapeta and add at least a day to make it 7 if you're going to start from 191. It wouldn't hurt to add an extra contingency day in case of bad weather and you get shut down up for the better part of a day. Of course, you could also train up like some of these ultra runners out there and do it in about 24 hours as has been done by others, but then you miss much of the scenery.
Also, as has been said, unless willing to pack an ice axe and some crampons and know how to use them properly, I wouldn't plan to attempt it until late July unless it has been an exceptionally dry & warm winter/spring. If it was an exceptionally wet and heavy winter/spring, then probably best to wait until mid august.
I'd really love to give this trail a go as well in the next few years, probably starting at Leidy. Here's a decent video of someone else's trek along the trail when you have a good hour to watch it.