I've done that drive a bunch of times. The Oregon coast is beautiful, but maybe not so much in March, as the climate is fairly similar to coastal BC (mild and rainy). If it's clear, it's nice though.
For the California coast, the entire thing is nice, but it does add some driving time if you do the whole thing. If you want the best bang for your driving time, the Big Sur coast/Monterey area is probably your best bet. It's the nicest part (IMO) and it only adds about 2 hrs of driving vs just staying on the I-5. If you have more time, you can keep going up the 1/101 north of San Francisco; not as nice in terms of coast, but you do get to drive through the Redwoods. But if you do that you're probably adding another day of driving.
For going down, the I-5 is pretty boring most of the way, but there are tons of places to stop for the dog and lots of pet-friendly places to stay. My main advice for that would be:
-check the weather in the mountain passes (Grants Pass -> Redding area); snow likely won't be an issue that time of year, but you never know.
-don't stay in Stockton
-try to time your drives through LA to avoid the worst of the traffic. 2 hrs from Palm Springs -> LA in light traffic can be more like 4 hours at busier times of the day. Same applies to Seattle which can be a nightmare to get through at the wrong time of day.
So in terms of itinerary, something like the following makes sense for a 3-day drive down:
Vancouver -> Eugene or Cottage Grove
Eugene -> Sacramento area
Sacramento -> Palm Springs
For the way back, it'd really depend where you want to go. And if you're not otherwise going out to LA, you might consider staying just past there so you can spend the day there; though you can't really go to any beaches in LA with a dog. There's one dog-friendly one in Malibu, but that's about it.
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