Proxima Centauri is approximately 4x1013 kilometres away from us (or 40 trillion, if you like). To cross it in 10 years (from the point of view of an outside observer) requires an average speed of ~0.423 c, or ~127000 km/s. Thing is that unless you can accelerate instantaneously, which you can, you need to work your way up to that speed, and because you've spent so much time below it your peak speed now needs to be so high that you can't possibly ignore relativistic effects. There's a useful web page that's been copied about a bit called The Relativistic Rocket, from which I will take some or more of the equations below.
Because you can't ignore relativity, the meaning of "10 years" has also kinda gone out of the window because the time experienced by the crew of the ship will be less than the time seen by someone at Sol or Proxima Centauri observing the ship.
To travel in a co-ordinate time of $t$ over a distance of $d$ accounting for relativistic effects, you'd need a continuous acceleration of $$a = {d\over{\left({t\over2}\right)^2 - \left({d\over 2c}\right)^2}}$$ which in this case will be ~.2 standard gravities. The proper time experienced by the ship's crew will be less... $T = \frac{c}{a}\sinh^{-1}\left( \frac{at}{c} \right)$ or ~7.13 years. You can rerun the numbers yourself if you wanted 10 years shipboard time!
Your peak speed at $h = \frac{t}{2}$ will be $$v = \frac{ah}{\sqrt{1 + (ah / c)^2}}$$ which in this case will be a fairly brisk .72c. I hope you've got good shielding technology.
The good old rocket equation tells us that than when the delta-V is much greater than the exhaust velocity of your rocket, the mass-ratio of your rocket becomes enormous, requiring ridiculous amounts of fuel. Plausible fusion rockets will have exhaust velocities of no more than 0.09 c, and even in that case the amount of useful oomph will be low because only a small proportion of the exhaust will be travelling at that speed. Naturally, the relativistic rocket equation is even less forgiving that the regular kind.
Relativistic delta-V is tricky, but there's another useful measure known as rapidity which is $r = \tanh^{-1}\left(\frac{v}{c}\right)$ where $v$ is your regular speed and $c$ is the speed of light. The peak rapidity of the ship will therefore be ~.902. The total delta-r of the flight will be twice this figure (to go from "stopped" to r=.902 and then back to zero again), and so the relevant relativistic rocket equation will be $$\exp \left( \frac{\Delta rc}{v_e} \right) = \frac{m_0}{m_1}$$ where $\Delta r$ is your total change in rapidity, $v_e$ is your exhaust velocity (say, 0.09c), $m_0$ is the initially fully fuelled mass of your starship and $m_1$ is the mass of your starship with empty fuel tanks. (if you were running some numbers yourself, you may find that $\tanh^{-1}$ is called something like atanh on whatever fancy calculator you're using)
This gives you an impractical mass ratio over over 2900 just to get up to top speed. If you didn't want to whiz past your target at nearly three quarters of the speed of light, you need an eye-watering mass ratio of over 37000. By comparison, Ariane 5 has a mass ratio of 40 which is one of the highest anyone has managed to achieve and it does that using staging. Even if you had many fusion stages it seems implausible that you'd hit your target mass ratio and stil lhave a useful payload mass.
I thought the fuel was no problem for a civilization able to mine the gas giants for it.
Sure, you can mine a whole load of 3He and make some pretty good fusion rockets, but it just ain't enough. If you want a rocket to get you to the speeds you're after, you need antimatter, and quite a lot of it.
For complicated reasons, the exhaust velocity of a beam-core antimatter rocket (sometimes called a pion rocket) is about .33c. For even more complicated reasons, the rocket equation doesn't quite work for antimatter rockets, so you need a different and much more complex one (some details here, scroll down a bit). The equivalent delta-V to the delta-r calculated above is $\Delta v = c \tanh(\Delta r)$, or about ~.947c, Throw that into the Frisbee delta-V equation, and out comes a mass ratio of a little over 33. If your ship had a dry mass of 100000 tonnes, you need somewhere in the region of 1.6 million tonnes of antimatter, plus the same amount again of regular matter (much easier to obtain and handle) for your reaction mass.
If your tech level was even higher, you might try to create an antimatter fuelled photon drive, maybe something like Winterberg's ambiplasma pinch graser. With an exhaust velocity of c, you'd have a positively sensible mass ratio... $\exp\left(\frac{aT}{c}\right) - 1$ (where $T$ is the proper time of the trip!) or about 5. You'd now need "only" 250000 tonnes of antimatter for you 100000 tonne spaceship.
How you'd go about creating such ridiculously large quantities of antimatter is outside of the scope of this answer, but it won't be easy.
Should I use another propulsion method or flight plan?
Firstly, you should not use a rocket. Trying to travel interstellar distances in a sensible period of time with anything other than a magical matter-to-energy conversion rocket or reactionless drive is a mug's game.
You should probably use a beamed propulsion system. Mass beam propulsion is probably the best type to use, with a particular favorite of mine being the late Jordin Kare's High-acceleration Micro-scale Laser Sails for Interstellar Propulsion (sometimes called "Sailbeam" for short).
Basically, you shoot very high velocity projectiles at your spaceship, which vaporises them and bounces them off a magnetic nozzle to provide acceleration. You can have a huge propulsion array left behind in the solar system using good old solar power to drive it, and have a much more svelte starship and no requirement for half a million tonnes of antimatter to remain well behaved for a decade. Mass beams also reduce the need for the sort of giant planet-frying laser arrays needed to push giant laser sail starships.
Ideally you want your boost phase acceleration to be as high as possible, whilst maintaining crew comfort and ship integrity. If you can manage more than 1G, that's great, because that keeps the boost phase short and reduces the need to accurately fire a relativistic machinegun across interstellar distances. You could build up acceleration to let crew and cargo acclimatise.
How to handle the problem of gravity onboard the ship if I don't use a constant 1G acceleration?
Spin "gravity" will be just fine. There are plenty of other places to read about it, but I'll link one of mine from this site: How fast can a ship with rotating habitats be accelerated?.
The key thing is that the design of a habitat which uses centrifugal forces to provide artificial gravity can be tailored to suit the external gravitational field or rocket acceleration, and a suitably designed hinged rotating section can provide a constant 1G internal artificial gravity for an external acceleration of anywhere between 0 and 1G inclusive.
I also suspect that you can have a lower-than-Earthlike acceleration on your ship without seriously ill effects, but you probably wouldn't want it any lower than the gravity of your destination planet (if there is one). You can also have a higher spun gravity, so you can do things like slowly reduce apparent gravity to ease acclimatisation after a >1G boost phase.
About what amount of fuel would the spaceship need? Just to get an idea. The ship is to host 10,000 people tightly packed (again for scenaristic reasons), so it's rather large.
If you're using mass-beam propulsion to accelerate you up to speed, you need no reaction mass for the boost phase of your flight. Now, it is possible to use a mass-beam system to decelerate too, but it is tricky and risky so maybe don't do that.
Instead, you can use a magnetic parachute.
This diagram is taken from The Magnetic Sail: Final Report to the NASA Institute of Advanced Concepts and shows that you can get a 1G deceleration rate without using a rocket at all, if your magnetic brake is good enough, and your target star has a strong enough stellar wind. The deceleration rate at Proxima Centauri is not something I'm going to compute for you right now, but it should be possible to perform a sort of "magnetocapture" manoever (the stellar wind equivalent of an aerocapture) to enter orbit around the target star, and your magnetic sail can then be used as a means of slow propulsion in-system.
What's left, then, is to carry enough fusion-fuel to run a reactor to power the magnetic deflector nozzle during the acceleration phase, the magnetic parachute during the deceleration phase, and life support and shipboard functions for the whole flight. You can then add enough fusion fuel and reaction mass to be able to use a fusion rocket to get around a bit more promptly in the target system, but honestly it might be better to carry some in-system transport spacecraft and treat the starship as a big space station instead.
A sensible mass ratio for a fusion spacecraft is probably somewhere between 1.2 and 2.5, depending on your patience and mission timescales (crawling through the Project Rho realistic fusion designs section for information on ships that are mostly designed by scientists and engineers aiming for plausibility). A thousand-tonne in-system spacecraft might, therefore, need somewhere between 200 and 1500 tonnes of fuel and reaction mass.
addition in response to comments
As far as scaling beam propulsion goes, that depends very much on the kind of beam you're using. A single big laser sail could be driven by a gigantic phased array, and you might be able to add new array elements as you wish, but that big sail could be incredibly large (a thousand kilometer across, and probably more!). A magnetic sail on the other hand can be made much larger much more easily, and it is possible to use multiple microsail launchers to target a very large spacecraft magnetic sail making it easier to scale the launch system "horizontally" as well as being able to use a smaller number of much more capable microsail drivers that push larger microsails, or push them for longer (so they get faster).
