Sword & Shield: Very fast, and the only weapon that lets you use items without having to sheathe it first, including not only healing but offensive items too, like the slinger. The shield isn't very good, so you mostly want to dodge things and only block in emergencies. While the raw damage isn't high, because it swings so much, it tends to be good at applying status effects. This is made even better when you use it in conjunction with status-inducing throwables (i.e. paralysis daggers) and slinger ammo. It also has an attack that makes it very easy to mount monsters, so you can knock it down for the rest of the group to hit it.
Dual Sword: Very fast and mobile. Pressing R2 activates Demon Mode, which lets you attack even faster. Demon Mode attacks can't bounce off the monster, so it does pretty considerable damage at the cost of stamina. You'll want to use this whenever you get a big opening (i.e. the monster is stunned or knocked down). Just be careful not to hit your teammates with it.
Longsword: Fairly fast and long reach, good for cutting off monster tails. Every normal attack builds your spirit meter and the R2 combo spends it. If you land the very last hit of the R2 combo, you gain a spirit level (your bar outline changes color). Each spirit level increases your base damage by a bit, so your main goal is to get to max level as soon as you can and stay there the entire fight. Your spirit level decays over time, and some attacks can spend spirit levels in exchange for damage.
Insect Glaive: This is actually a 2-part weapon. You have your glaive and your kinsect. The glaive is a fairly straightforward fast weapon. The kinsect is what complicates things a bit. You can send your kinsect flying at monsters when your glaive is drawn, by aiming and firing it like a sling. When it hits a monster, it will change color. If you call it back when it has that color, you get a buff. There's three possible buffs - red, orange and white - and which one you get depends on which part of the monster the kinsect hit. White is speed, Orange is defense and Red is attack. The most important thing is to keep Red up, because it not only makes you do more damage, but also totally changes your attack animation to a much better moveset. You can also use the glaive to pole-vault on monsters backs, making it very easy to mount them.
Lance: Offensively, lance is very simple - you just poke things to death with either attack button. There aren't any complex combos, modes or offensive mechanics. The main benefit of lance is you have a shield which can block almost any attack while taking little to no damage. The dodge is also a very fast short hop that has a lot less recovery than the other weapons' dodge roll, and tons of invincibility. The lance can also attack while blocking and dash while blocking. When you put that all together, it's a very very safe weapon, so your goal is to learn the monsters attack patterns and be constantly attacking, block/dodge as it's about to hit you, then immediately go back to attacking again. It's a good weapon for fighting dangerous monsters safely. While the moveset is simple, the complexity comes in learning how to fight monsters efficiently with it.
Greatsword: Greatsword swings slowly and you move extremely slowly with it drawn. Because of this, most of the time you'll want to unsheath it, swing once or twice, roll, then sheathe it again. Your big damage with GS comes from its charged attacks, which get stronger the longer you charge them. They're very slow but hit extremely hard when fully charged, so with every monster your goal is to find out when you can safely charge to land the big hits. You can also cancel the charging animation into a fast shoulder tackle, if you find you misjudged the attack timing. Because of high damage, long range and slashing damage, it's one of the best weapons at cutting off monster tails.
Hammer: Hammer is very slow and has short range, so you need to be very careful about when you attack. If you hit a monster's head with it repeatedly, the monster will be temporarily knocked out, and your whole group can wail on it for a while. You can charge some of its attacks for more damage.
Hunting Horn: Hunting Horn is an offensive support weapon. Every attack with it gives you a musical note, and specific combinations of notes lets you play a song with the R2 attack. Each horn has a different set of songs, and playing them gives your entire group a buff. Because songs require specific combinations of notes, you have to pay close attention to not only when you attack, but which buttons you're using to attack with, and in what order. If you play a song twice, the effect becomes stronger, so you'll be spending a lot of time keeping buffs up. It's a striking weapon like the Hammer, and it has very good range, so it's also good for hitting monsters heads and knocking them out. Even though it's a support weapon, it still can do decent damage, so you'll want to balance doing damage and keeping the group buffed. Despite being so big, its movement and attack speed is actually quite fast for a 'large' weapon.
Gunlance: This is a heavier, much more offensive version of the Lance. You lose most of its defensive actions like dashing or attacking while blocking, as well at the Lance's fast attack speed in exchange for a lot more damage. The Gunlance shell attacks ignore armor and always hit for full damage, but consume ammo which has to be reloaded. You can aim your shots by using one of the Gunlance's melee attacks, then firing a shell while the lance is still sticking out. Because of this, its size and the shell's explosives, it's notoriously good at breaking specific monster parts that other weapons might find hard to reach. Several of the Gunlance's attacks do very high damage, but either require ammo or have a long cooldown associated with them. Another drawback is all of its strong attacks consume sharpness very quickly. This makes resource management a very big part of maximizing Gunlance damage. Even though it's not as defensive as a Lance, the shield is still very strong at blocking so you can play defensively when you need to.
Switch Axe: A weapon with a fast Axe mode and a slower Sword mode. It has a lot of range and the sword mode does very high damage, while having attacks that can't bounce off the monster. Most of its attacks chain into each other, so you can keep a combo going forever, so long as you have enough stamina and don't get hit. The sword mode is limited by your sword meter (the blue bar under your sharpness meter). Every attack with the sword spends some of it, and every attack with the axe gives some of it back. It also regenerates over time while in axe mode. If you run out of meter, you'll stop attacking and the sword will forcefully switch back to axe. You can use R2 during a combo to attack and change mode at the same time, so as long as you have stamina and pay attention to your sword meter, you can stay constantly attacking.
Charge Blade: This is probably the most complex weapon. It has a fast sword+shield and a slower axe mode. Attacks with the sword build up phial energy. Phials are the little white bottles under your sharpness meter. When you have enough (the phial indicator will be red), you can store it by pressing R2+O and it will refill your phials. Your strongest attack is the Axe mode's Elemental Discharge, which requires phials to use. Your goal with this weapon will be to use the sword to charge phials, switch to Axe and spend your phials with Discharge for big damage, then switch back to sword and repeat. Random trivia: in the JP version it's called 'Charge Axe', because you're charging energy to use with the Axe. For whatever reason, the English versions renamed it to Charge Blade.
Bow: A light, very fast ranged weapon. It's kind of a ranged version of the sword and shield. It has good mobility and attack speed, but individual shots do little damage, so it's best used with elemental damage and status effects, aimed at monsters' weak spots. Bows also have temporary buffs called 'Coatings', which give your arrows various special properties. The Dragon Piercer shot does more damage the more parts of the monster it hits, and it can do quite a lot if you aim it so that it hits the head and goes out the tail. Each bow has an ideal firing range called the 'Critical Distance'. Shot fired outside the Critical Distance do significantly less damage. If you're in the right distance, your targeting reticle will have a second bright orange circle inside of it.
Bowgun: The is the slower, heavier ranged weapon. You can't do this in the demo, but normally you customize your Bowgun with different parts. Which parts you use determine whether it's a Light or Heavy Bowgun. It generally hits harder than the bow, so it's more effective at straight physical damage, though you can still use it well with elemental damage or status effects. A lot of Bowgun use is matching your Bowgun parts, armor skills and ammo type to the specific monster you're fighting. It's much slower than the bow, so dodging with it is quite a bit harder. You'll want to make sure you're far enough away from the monster to give you time to react and get out of the way when it attacks. Like bows, Bowguns also have a Critical Distance.