Archdemons are the
final boss of each area/Archstone, so until you have killed at least one of them in another archstone, you can't progress to the next part of Boletaria (1-3). There's no immediate rush to do that, the nice thing about Demon's Souls is you can tackle the areas in any order you like.
There's no real order, but going left to right is a standard enough path - we number the archstones and areas left to right so Boletaria is area 1, so the first part upto Phallanx is level 1-1, the dragon bridge up to Tower Knight is 1-2, the first area of Stonefang tunnel is 2-1, and so on. You'll see this system referenced a lot.
http://demonssouls.wikidot.com/stats
http://demonssouls.wikidot.com/weapons
The wikidot Souls wikis are my favourite, but there's a few others. All the wikis tend to list the weapons with their lore, location, stats etc so they're going to be a bit spoilery, but if you take a look at the general overall weapons/stats pages they will explain what all the numbers and symbols mean without spoiling anything else.
Basically when you level up, you want to level up the stats that will give your offense the biggest stat bonus on the weapon you're using. All weapons gain a damage output bonus based on your stats, and each weapon has a letter grading on what stats it scales with. That' the letter/symbol combintion on each weapons stat page. For example a big heavy weapon will probably scale well (grade B or above) with Strength, so leveling up strength will increase the damage bonus you get when using that weapon.
People tend to pick a weapon or two (or type of weapon) and stick with them throughout the game then level up the stats it scales well with. Same for magic, if you're using magic putting points into magic/int will make your spells more powerful or allow you equip more spells.
The consumable soul items (unknown heros soul etc) are to be used whenever you like, they have no other purpose. Use them to level up, upgrade weapons etc. Boss souls DO have another purpose, they can be used to craft unique weapons or exchange for spells, and each boss soul may have more than one use, so hang on to them until you're sure you know what you want to use it for (if you don't want to use it for anything else, consume it for souls). The boss souls you might want to look up on the wiki as the game isn't great about telling you before hand what each one does, especially for weapons.
World/Character Tendency: Probably the most esoteric and initially confusing aspect of Demon's Souls. There are unique events, enemies and characters in each world that happens only when your character has character/world tendency at either end of the scale (pure black, pure white). Being in black tendency also effects item drop rate and strength of enemies.
The basics are - Kill a boss in an area, it notches that areas world tendency up towards white, die in human form in that area and it shifts it to black. So if you want to go all the way white, kill bosses. If all the way black, use ephemeral eyes (or defeat a boss) to return to human form (full health bar) and die in that area.
The tendency of an area won't take effect until you return to the Nexus. This is why you see a lot of bloodstains in the Nexus, after killing a boss and returning human, a lot of players will go to the Nexus, fall from a high place to die in the Nexus so when they return to a world they are in spirit form and dying there won't affect their world tendency.
Charater Tendency can be changed similarly with killing NPCs (goes black) and towards white by killing human invaders or black phantoms (NPC invaders).
http://demonssouls.wikidot.com/world-tendency
http://demonssouls.wikidot.com/character-tendency
It's initially confusing and not something I would recommend trying to achieve 100% on your first playthrough, think of it as extra content for NG+ or another run through.
Good luck and if you need more help ask the experts in this thread (I'm not one of them!), we'll all keep it spoiler free, that first playthrough is special!