Bear in mind that if you really accelerated to light speed on a trip to Mars, circa 150 millions miles, the acceleration would be awesome. Assume you accelerate so that you reach light speed half way there, and then decelerate for the rest of the trip. That would require an acceleration of 230 miles per second per second, or 1.2 million feet per second per second, or 38,000 g's! Everyone and everything on board would be crushed the instant you turned on the engines. Well, maybe you're assuming that if we had the means to travel at light speed, we'd also have some way to cancel the effects of the acceleration.
My first question is: Why do you need to be able to travel around the solar system in minutes? Slow it down to days and it's still a trip that people could make routinely, but it would be painfully slow for interstellar travel. Like if it takes, say, 2 days to get to Mars, that's an average speed of 3.1 million miles per hour. Incredible speed! But at that rate it would take you over 200 years to reach Alpha Centauri. I think the easy answer to your problem is to just slow the ships down to the point that interplanetary travel takes reasonable amounts of time but interstellar is multiple lifetimes.
If you really need your ships to reach another planet in minutes for your story to work, I guess you'd have to posit some reason why they either, (a) can't travel far from the Sun or a planet; or (b) can't travel more than x distance.
RE (a) Maybe they are solar powered. Too far from the Sun, not enough energy, they don't work. Hmm, but why couldn't you put a nuclear reactor on board to supply energy instead? Maybe some hand-waving there. Maybe they need a solar wind for some reason?
(b) is easier: They require huge amounts of fuel. The longer the trip, the more fuel, of course, But it doesn't go up linearly, it goes up exponentially, because for a long trip, you have to carry enough fuel to propel all the fuel for the rest of the trip. Just like present-day rockets require huge amounts of fuel because you have to burn enough fuel in the first few seconds to get the fuel for the entire rest of the trip off the launch pad, etc. You could say that the fuel requirements rise such that for a trip of light-years, the equation becomes unsolvable, an infinite amount of fuel would be required. Ok, I haven't worked out the math if that's actually possible, but at least, an amount of fuel so large that it would take all the fuel you could possibly produce for hundreds of years.