I would personally be excited to have such a trip in my immediate future, and would be planning away right now. That would add a couple of the western most states to my log book, and some low, slow sightseeing over country that I have seen from commercial airliners, cars, and trains.
Small hints about a flex day are an understatement. Weather on that many days will have at least one bad day.
Even if your instructor is instrument rated, there will be a day that is not suitable for a Piper 140. That is the thunderstorm and spring rain time of year. The long term forecast when you buy your airline tickets on the East coast will not be the weather you fly in. Wind from 3/4ths of the compass are net headwinds, as Ed Haywood advised. 800 hours of cross country has taught me the same.
Remember, weather travels from the West to East, if you are behind a front, you cannot progress any faster than it does. That is a trade off for the predominately easterly flow of air that provides a tail wind for your trip.
With a new to you plane, you are likely to find problems that need to be addressed enroute, and as anyone who is following mechanical complaints here, they often can not be corrected in the same day found.
When time is short, poor choices are made, so planning for a 9 day week, and being pleasantly surprised at 3 days is the way to go.
Now, after all that negativity, I am envious of your proposed adventure. The time of year and states that you will cover will be unforgettable, as long as you are not in a time constraint. My long days are in 3 hour sections, and up to 3 per day works. Two people in a dual control plane is ideal, and makes the long days go smoother, less fatigue. Fuel stops take a minimum of half an hour.
Take a camera, and take a lot of pictures. I have flown over most of the states east of the Rockies, and particularly in the spring, all are interesting, most are beautiful.
In the early part of the trip, plan relatively large airports, so there will be repair facilities on the field (Pessimism pays when flying a new to you plane). After the "New" wears of your used plane, and you have some hours on it, using smaller airports in the planning is just fine.
Get east of the Rockies, then plot the great circle route, and keep as close to it as you can, but do not consider it as a requirement, just a time saver. Weather and most suitable airports are the more important deciding factors for a vacation such as this.
Deviate for the most interesting places to fly across, the Meteor Crater is an example of a landmark best seen from the air.
Starting with plenty of time AVAILABLE is the key to this trip being memorable in a positive way.
