I'm having trouble figuring out what equipment is worth upgrading, mainly weapon.
What type of build are you going for and how in depth are you planning on playing the game? If you are going to go after the platinum it is a very different approach then just trying to play through the game once.
If you are focusing on staying light in terms of character tendency, Demonbrandt is probably the most powerful weapon you can reasonably attain somewhat early in the game. You don't have to worry about upgrading it, you won't do anything with it until end game when you combine it with Soulbrandt to make Northern Regalia.
For general useage without any real weaknesses, Large Sword of Searching is good for a first playthrough.
http://demonssouls.wikidot.com/large-sword-of-searching
Work on upgrading one of the weapons listed to what is required and then stick with that until you kill the boss to get the required soul to upgrade it(Scimitar to +8 is probably the easiest and most versatile overall out of those). This isn't a very good PVP weapon at all, but for PVE particularly on your first play through it is very effective(and doubles your luck rating in terms of drops which can be handy).
For ranged I'd say Lava Bow hands down for the first play through-
http://demonssouls.wikidot.com/lava-bow
Get any of the bows listed to +7 and as soon as you beat Flame Lurker(2-2 boss) you'll be able to get it(you already killed Armor Spider so you should have his soul). In later play throughs when you starts start getting a lot higher there may be better options, but I honestly don't think anything is close to Lava Bow on your first play through.
For caster weapons the best is the Talisman of Beasts, but that is hard to get. Until you can get that, the plain wooden catalyst is the best overall. The Silver Catalyst gives you more ~20% more MP, but reduces the power of your spells. Wooden Catalyst has damage close to the Insanity Catalyst but doesn't reduce your MP by 50% like the Insanity Catalyst. If you are rolling faith, you are stuck with Talisman of God until you can get the Talisman of Beasts.
For shields what you are going to want to use will change around a lot depending on what you are using. For PVE content, most of the time I use Adjudicators(4-1 behind a red eyed skeleton- be careful with them) as the constant healing effect is handy. For certain areas, 3-1 and 4-2 come to mind, the Dark Silver Shield is very nice(5-3), particularly for melee based players. The Tower Shield is really nice, but insanely heavy making Endurance demands too high to realisticly use on your first playthrough unless you are going for a straight tank(no real rolling at all).
Unless you are going for the platinum trophy, I wouldn't worry too much about upgrading the 'wrong' weapon. The really powerful weapons often can't be upgraded or can only be upgraded using Colourless Demon's souls which are very rare(at which point you should have figued out for yourself what you want the most). Boss souls are only used for one weapon upgrade each, so any upgrade available via a boss soul isn't going to hurt any other upgrade you may want to do. Most of the shard upgrades are more for situational use(outside of Crescent weapons which any heavy caster is always going to want to have, but they don't carry them for the melee).
