The maximum distance between Earth and Mars is ~401.3 million kilometres. Leaving aside the pesky star in the middle for now, to cover this in 30 of your Earth days is going to take a fearsome rocket.
You'll be needing a continuous thrust rocket, where the engine has a steady acceleration up to the half-way point of the journey, at which point the rocket flips over and then does a stead deceleration burn to the destination. The brachistochrone equation can tell us what sort of oomph you need: given a distance $d$ of 401.3 million km and a time $t$ of 30 days, you end up with an acceleration $a = \frac{4d}{t^2}$ of about 0.24m/s2. That's not obviously a whole lot, but it adds up over 15 days of boosting to get you a peak speed over 300km/s which is pretty formidable.
With a required delta-V of ~620km/s, you're going to need a potent rocket engine, or a rather silly amount of fuel. Lets say we have a generous mass ratio $R$ of 10... that is to say, the loaded spacecraft masses 10 times as much fully fuelled as it does when its tanks are dry (or alternatively, the fuel masses 9 times more than the empty ship). You can use the specific impulse formula to work out that you need a rocket with a specific impulse of 27426 seconds, or an exhaust velocity of 268km/s. This is at the very top end of what a VASIMR might produce.
Lets say your spacecraft weighs about a 200 tonnes with full cargo but no fuel. That gives it a "launch" mass of 2000 tonnes. I put "launch" in quotes because I'll handwave this as leaving from high Earth orbit. To reach the required acceleration, you need an initial rocket thrust of nearly half a meganewton. With the required exhaust velocity, that means a rocket power of about 64 gigawatts. Clearly, no VASMIR is going to manage that, as it would require a giant nuclear reactor to operate. Clearly, you need a nuclear drive. I can't find any fission designs that are this efficient and powerful, so you'll be wanting fusion.
The ever useful Project Rho suggests two research papers on fusion driven spacecraft designs... this makes the designs plausible, if not yet actually possible.
- Realizing "2001: A Space Odyssey": Piloted Spherical Torus Nuclear Fusion Propulsion
- Z-Pinch Pulsed Plasma Propulsion Technology Development ("HOPE Z-Pinch")
Both these designs have much lower mass ratios than your ship needs, but they should give you a starting point. Critically, both ships have a fairly similar layout, showing quite how much space needs to be given over to heat sinks for your monstrously powerful nuclear engine:
Your delivery ship will have much more fuel, and a substantially larger engine because the Hope and the Discovery II are plausible ships with sensible milligee accelerations, and your nuclear monstrosity needs 10 to 15 times what they have.
Note that if you relaxed your shipping requirements slightly to be "30 days at closest point" (which gives you far fewer launch windows) you can use either of those ship designs as is for your needs. Only for the "30 days regardless" requirement do you need crazy rockets that carry ten times the fuel and have ten times the acceleration.
For the shortest possible trip, your rocket could make it in a little over 11 days with a mass ratio of more like 2.33, meaning it could carry several times as much cargo.
how much does it cost to use
Well, the trite answer is "as little as is required".
You do need to be able to put a few hundred tonnes of cargo into space cheaply (or manufacture everything you need out there) and you also need to be able to produce several hundred tonnes of refined nuclear fuel (requiring deuterium separation, tritium generation, lithium separation, etc) and get that up to the relevant orbit, too. You need to be able to do the same job at the Mars end of the journey, or your delta-V budget quadruples. One assumes that if you can't refine fuel in Mars orbit, you send it there via slow, infrequent but vastly cheaper and simpler Hohmann transfer.
The ship will need work after every flight. There will be micrometeor damage, radiation damage to the drive nozzle, neutron-activated drive components to replace and decontaminate, whatever else.
Clearly you are talking about a society which can do this, because flying to Mars in a month, on demand, regardless of its current orbital location with respect to Earth is a very technologically challenging, as I hope I've made clear.
what materials would be valuabe [sic] enough on a martian colony that it would be worth that cost?
Any piece of critical equipment in an emergency. Nuclear reactor parts, medical equipment, etc. Most likely it will be people that are the critical resource, and more importantly they're also the thing that suffers the most in long duration spaceflight.
A sensible colony would have backups of all the critical things, and replacement stocks would be kept topped up via slow but cheap(er) freight.
Please consider acceleration and decelleration [sic] of the material
A quarter of a tenth of a gravity. Don't you worry. Your ornate glasswear will be just fine.
Of course, landing stuff on the surface of Mars might be a bit more, uh, sporty. But clearly you can land people on there, so as long as whatever you're sending is no more squishy and fragile than J. Random Meatbag, it'll be fine.
It's not nessecary, but I would like the solution to involve a railgun, b/c they are cool.
With a max velocity at flipover of more than 300km/s, I simply don't know how you'd make a railgun that could throw a projectile fast enough to be of any use at all. Sorry about that. Some fusion reactor designs use railguns to throw plasmoids, but that's an internal detail of the reactor and not the sort of cool that you can really look at or interact with externally.
