Multiplayer isn't required to play it; but I really do enjoy it more with the internet connection active. Even if I don't actually play with other people, I seem to subconsciously (and consciously, to an extent) enjoy the game a lot more when I am constantly running past player messages/hints, and the blood-stains of players, as well as seeing "ghosts" of other players doing things all around me.
On-topic: I'm not sure it really matters which you play first. Demon's Souls DID look a SLIGHT bit better graphics-wise, but I'm pretty sure that's because of two things. One: the game was quite a bit smaller, over-all. And two: the levels were sectioned off - by which I mean, they were separated by loading screens. Whereas Dark Souls seems to be all one huge, preloaded game.
In addition, Demon's Souls was SLIGHTLY harder, at least in comparison to my playthrough of Dark Souls so far. Mainly because dying there cut your health in half (or down to 3/4, depending..) and also turned your world tendency black - making the enemies even stronger than they were before you died. So you get triple screwed; you lose your souls, half the health, and stronger enemies.
Oh, and as I read this thread while I'm typing: as others have said. It's God Hand hard (though I haven't actually played that one.. heard it's fun though) and not fake difficulty, like IWBTG. There are very few times in either Demons or Dark, where you will blame a death on the game instead of your own over-eagerness, impatience, or lack of understanding in regards to a boss, or a trap, or something of that nature. That said, however, I have run into one part so far in Dark Souls that infuriated me. Sure, it was still my fault, but it was just a lame death all-around.
*Semi-Spoiler, but not really*
At one point in the game, you are in a swamp, when you come across what looks essentially like a water-wheel from a mill or something, but is also an elevator. And at the top, you clamber off it and onto the next level, to continue on your way. Unfortunately, there IS a gap, so you have to roll/jump - and my timing was less than spectacular. So I fell to my death. It just felt like the one cheap death in a game full of deaths normally caused by my own stupidity.
Also, on Eldermage's comment about a load carrying stat: he is correct that there is no limit to what you can carry in your "back-pack" or whatever extra-dimensional space carries the loot. But there is still a stat regarding how much you can wear.
Anyway: tl;dr - either one is a fine jumping off point. Demon's Souls is slightly harder in some ways.