So typically you have a hub where you organise/buy/sell your items, have gear crafted from materials you've gathered both from the environments and the monsters you've killed, eat food for temporary buffs and so on. You accept quests from a list of available ones at a kind of guild, at first they'll be really simple things like kill a few of the boars you've probably seen roaming around or gather a few plants and make some health potions, but soon enough they almost exclusively become hunting the bigger monsters like the ones in the demo (and usually a few of them will be much, much larger). When you leave on a quest you'll appear in one of the game's maps (usually there's around 5 large maps, some arenas and a few smaller maps for fighting very specific monsters) and it'll be pretty much the same as the demo - find the monster and kill or capture it. Often there are optional secondary objectives like kill ten of the smaller monsters accompanying the main one or bring a specific item back to camp or break a specific part of the monster, and you'll get more rewards for completing these. Sometimes you'll get quests to hunt multiple monsters. I think most people are expecting around 25-30 large monsters in the game, with the possibility of more being added as DLC later.
Your main reason for progressing is usually to make new armour and weapons with new elemental resistances, different abilities, and strengthening/upgrading them. You do this by using materials you gather primarily from killed/captured monsters - some materials you pick up off the floor during fights when you break part of the monster, some you get from carving the monster when you kill it, some you get as a reward at the end of the quest (and you get an increased number of you capture rather than kill, but then you don't get to carve). You then use this new and improved armour and weaponry to take on even stronger monsters. Some rare materials will need a few kills of the same monster for you to be lucky enough to see one, or require that you break a certain part of a monster during the hunt for it to have a chance to appear.
Hope that helps!
