You should plan on taking the fastest (shortest) route - with the idea of checking the weather and road conditions at the time for possible modification. The fastest route is:

I-84/I-80/NE-2/I-29/I-70/I-64/I-57/I-24/I-75/I-20/I-26. This is right around 2925 miles, and 6 days is recommended.

Notes:
1. Take I-435 northeast around KC.
2. Take I-270 north of STL, then I-255 south to I-64.
3. Take I-285 northeast around ATL.
4. Sorry to take you through Atlanta, but I-40 is closed at the TN/NC state line.
5. You can cut off some miles right near the end by taking US-178 and US-301 from I-20 to I-26.

If you go down I-5 to I-40, you will be adding 400 miles to your trip, and an additional day. If you go to I-10, that adds yet another 200 miles.

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