The Drake Sword is not recommended in the long run because its damage does not scale with any stats; this means that while it does an awesome 200 damage in the early game, that's not much compared to the damage you can do with high stats and a weapon with good stat scaling.
I personally moved on to Astora's Straight Sword, as recommended by the Wiki; you can find it in The Valley of Drakes, guarded by an undead dragon hanging off the edge of a cliff. If you can pick up two pieces of Twinkling Titianite and upgrade the sword to level 2, it'll already be doing more damage than the Drake Sword* . It's also very forgiving in terms of stats; it scales equally with all of Faith, Dexterity and Strength, so you can make some missteps in your stat distribution without suffering too much for it.
However, as the name implies, it's a straight sword, not a curved sword like you wanted. If that's what you're set on using, the easiest one to get at this point is carried by the merchant Shiva of the East, who is unlocked by joining the Forest Hunter Covenant. He carries the Shotel, which has the nice side-effect of ignoring shields, and a couple of katanas you might want to try out (though they're not curved swords, strictly speaking).
Otherwise, the skeletons in the Catacombs (and not the Firelink Shrine graveyard, unfortunately) sometimes drop Falchions and Scimitars; you'll probably go to the Catacombs soon after Quelaag's Domain, though if you want to you could sneak in there right now.
There's two things to keep in mind when switching, though: these swords scale best with Dexterity, so if you haven't been pumping that stat you might want to start doing so now; also, none of them will do as much damage as your your Drake Sword - although they scale with stats, they start at 82 base damage, compared to the Drake Sword's 200. You probably won't want to switch to one of the curved swords for new areas until you've put some upgrades and practice into it.
If you decide to stick with curved swords throughout the game, I recommend picking a particular one and upgrading it to +10; after that, you can transform it into Quelaag's Furysword, which is one of the best swords in the game.
*The wiki says level 3, but that's the point at which it does more base damage; the scaled damage will eclipse the Drake Sword earlier, though of course this depends on your stats