My main character still has her first Rusty Iron Bow! It's way more useful now that she can actually do something with it... Anyways, it's the original one from the dungeons. And it's just on the wall of my house, even though I hardly ever use it. Nostalgic purposes, I guess.
Back to the point... This is actually kind of a hard question. My characters all use bows as their main weapon type, so you would think I have some kind of plan for how I move from bow to bow. Basically, I guess I just use whatever happens to be the best bow I can find. I'm ashamed to admit I almost never Enchant anything, but when I do it's always Ebony or some special bows that look cool, like Perfect Amber. I don't even bother with Daedric bows anymore, I've grown bored of them over the years. They're too easy to get (to a Morrowind player it seems sooo wrong that you can get Daedric off of every other bandit that you come across, and I still haven't gotten used to it after all these years of Oblivion...) and not as pretty as Ebony, for example.
I go bandit hunting and kill the archer bandits before they can use up all their arrows or break their bows and see if there's anything better than what I have now. My higher level characters use Ruin's Edge (it's pretty!) and self-Enchanted Ebony and Amber bows, my midlevel characters usually use Shadowhunt. I have some mods that add really cool bows, so I use those randomly too.