That's pretty much impossible. There's mountains in every direction from Seattle.
If you are willing to go across Snoqualmie which looks to have chain law up, I would highly advise going to Salt Lake City, UT...and then scoping out conditions from there.
If Wyoming isn't too bad cross i80 and head down i25 from Cheyenne, head East on i70, then take 287 down to Amarillo. This is the shortest route I believe, at a cursory glance.
If Wyoming is gonna be closed down or what have you, you can cut down i15 from Salt Lake City and take us-6 down to i70 East, then us191 south through Moab, to us491 down through cortez and into New Mexico down to i40. I have done this in winter, though those areas can get bad weather too. This will also add some mileage, but not too egregious (better than sitting in Wyoming for days).
Cutting down to i70 and crossing through the Rockies to Denver is also an option.
Trying to head south to Cali is gonna add around 400 miles. But basically jump on i5 all the way down, then i210 around Los Angeles to i10 East. Definitely some mountains going through Oregon that way, too.
You could also run i5 down to Portland and then run i84 across and down to SLC, if you don't want to run i90 over to i82 in Washington. That will add 100 miles. Not quite as bad as running down to Cali and i84 looks pretty good across Oregon right now, even through the blue mountains (Pendleton OR to Baker City OR).
Once you are at Salt Lake City follow my advice above.
Cutting down to i70 and crossing through the Rockies to Denver is also an option.
Really bad idea. I-70 west of Denver is almost always a mess in the winter. ( read as will have to chain up almost certainly)
No way around the mountains. My suggestion is to get each states 511 apps and see which has the best weather.
Above all else, if you are uncomfortable, don't drive it.
Running team we have run primarily coast to coast runs, and many times in winter we see i80 is shut down in Wyoming or is going to be and have run i76 down to Denver and ran across i70.
It certainly has the potential to get bad there but it can be perfectly clear even in the winter.
Panama canal, just saying...lol
Try this route, avoids most mountains and no snow (except for Seattle itself) right now: https://goo.gl/2Er4KP (The link feature wouldn't recognize either the long link or the shortened link for some reason :/)
Trying to head south to Cali is gonna add around 400 miles. But basically jump on i5 all the way down, then i210 around Los Angeles to i10 East. Definitely some mountains going through Oregon that way, too.
If you're going to go the long scenic route through Cali, you're better off going down I-5 to 120, over to 99, down to 58, across 58 to I-15, jumping on I-40 in Barstow, then taking US-93 south from Kingman into Phoenix and picking up I-10 from there.
You'll still have to deal with mountains (as was previously mentioned there is no way out of Seattle that doesn't involve mountains), but the only really hairy and potentially snowy ones will be in southern Oregon and northern Cali. Once you get south of Redding, you should be in the clear as far as weather goes.
Can't help but wonder how did the OP come to be in Seattle without encountering mountains?
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I am new. Could someone advise me as to the best interstates to take to drive from Seattle to New Orleans, and avoid snow, ice, and mountains? I would also like to avoid Los Angeles if at all possible. Any advice will help! Thanks! - Sue
Interstate:
Commercial trade, business, movement of goods or money, or transportation from one state to another, regulated by the Federal Department Of Transportation (DOT).