We actually have nuclear tech that can propel a spaceship to sustained acceleration rates of 1G or higher.
The problem is that, by treaty, this tech can't be used because it involves nuclear explosions— very small, shaped nuclear explosions. But by the Outer Space Treaty and the Test Ban Treaty any such nuclear explosions are forbidden.
Look up Project Orion. We almost built the thing back in the ‘60s. Still could.
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There's also NERVA, which is also viable:
en.wikipedia.org/wiki/NERVA
And Project Timberwind:
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And DRACO:
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As it is, SpaceX has already reduced cost to orbit to $1,200 USD per pound. Starship, when completed, is expected to reduce the cost to under $200 USD per pound. These rates are achieved to a great extent through reusability of the rocket components— it saves a lot of dough when you're not having to build an entirely new rocket every time you want to launch something into space.
But even with that, chemical propulsion will always be the "slow way."
As for interplanetary and even interstellar travel, solar sails might be a viable solution— there's no fuel needed, just the "sail" and the light of the Sun. Constant acceleration to midpoint,* then shift the sail against orbital movement, and decelerate constantly to the destination. For manned interplanetary travel, it's probably not really viable past Jupiter, as the rate of acceleration obviously falls with distance from the Sun. Still, it would be exceedingly cheap, and possible to launch supply "trains" as it were, essentially looping in a constant series of craft which link up with payloads (possibly even manned) in Earth orbit (or nearby), and then "dropped off" at a target destination, such as a Mars colony. The sails never land; they just loop and loop and loop, with the payload being boosted with chemical (or nuclear) rockets to connect with the sail at its Earth end, and if necessary collected by similar rockets in Mars orbit… or just dropped to the planet's surface via aero braking and either local rocket landers or parachutes. And stuff from Mars could be sent back the same way, making interplanetary trade indeed a viable possibility.
*The "midpoint" for such a trip would not actually be halfway; it would be to a point where solar acceleration would be sufficient to slow the craft to a viable orbital insertion point at Mars. As noted, this would be based on the amount of light and thus light pressure the sail would receive as the distance from the Sun decreases (or increases for a return flight.)