So I've got a titanite slab and a titanite chunk -- should I feed these to the serpent on hold on to them? I've already got my weapons set -- Lightning Iaito +2, Claymore +10, and lightning Spear +2.. Also, what would be the best route to take with my Claymore? I'm not on NG+
Keep them: feeding him titanite slabs or chunks will only give you large titanite shards in return. Titanite slabs are pretty rare in the game, so I'd really hold on to the one you have.
As for your claymore, it depends if you want to keep leveling and go the stats route, or stick to fire/lightning weapons (since they don't scale with stats).