I don't generally make alchemy hunting trips, but I scour the roads on my way wherever I'm going. I also buy up every ingredient the alchemists have, usually trading off dragon bones and scales, and whatever useless potions I acquire along the way. Or, I'll experiment until I find a nice expensive potion, make as many as I can, and then sell them to the alchemist in return for ingredients.
If you have Hearthfire, a garden helps enormously. I recommend planting canis root, imp stool, swamp fungal pod, snowberries, glowing mushrooms, blisterwort. That will give you the growable ingredients for some useful potions, like paralysis, smithing, restoration, and enchanting. You'll need to find salt piles, blue butterfly wings, and either Abecian longfin or Cyrodilic Spadetails though. For those, I usually give the skooma addict a healing potion, at which point the fish barrels along the docks become take, rather than steal.
Also, ask Farengar in Dragon's Reach, if he's the only Wizard. He'll ask you to take something to Arcadia's cauldron, after which she will give you a lot of ingredients. (There's also an alchemy skill book on her desk you can read, but not take)
The "fun" of alchemy isn't really in doing it. That's the work. The fun is in the result, having legendary armor, or that really kickass paralysis potion.