There are really two questions here, based around the two attributes of travel you've specified:
As has been pointed out in the comments, 10 million miles a minute is still below the speed of light, so it is theoretically achievable.
You'd need an enormous quantity of propellant and the ability to throw it out the back of your ship in a huge hurry, but that's all doable, if hideously expensive. The major problem is one of inertia. If you accelerate from 0 to .9c in an hour, any humans in your spaceship will be laminated against the back wall. If we assume that humans can withstand 5 Gs of acceleration indefinitely, it'd take 300 days of 5G acceleration to get to your proposed speed. So your warp drive needs to effectively break inertia. "Inertial dampers" are a key component of most space-operas, and they are pointedly never explained, because there's no realistic way we know of that they would work.
Probably, though, you want to do all this without carrying a tank of propellant the size of Ceres and a bottomless supply of energy. So the better bet would be something like an alcubierre drive, where your ship never needs to accelerate substantially at all, because the distance they're travelling has been modified. In this case, the only law of physics you have to be able to violate involves the creation of negative mass. Now, what that might do to things in the solar system is, at this point, entirely theoretical.
Really, What You Want...
Is the starship Heart of Gold, powered by the Infinite Improbability Drive. Then you don't have to worry about the laws of physics, as you need only determine how improbable a given course of events are, and the drive will do the rest!
Edited to add: "Practical and Speedy" space travel is, as far as the current laws of physics are concerned, not a thing. You have to adjust so many of them (or avoid them) that space travel in pretty much all space opera is just magic. It is necessary magic, but magic nonetheless. Even the Expanse, as good as it is, is a little light on where the energy comes from to provide the enormous thrusts its passengers have to deal with.