Tunifus is in the downstairs room of the Imperial Cult shrine in Ebonheart. Once you join the Imperial Cult, you should be able to purchase a "Divine Intervention" spell, which will take you to the door of the shrine from just about anywhere in the Vivec area (which is where you'll be doing most of Tunifus' quests, anyway. Note that the Vivec Silt Strider and boat platforms are far enough north of the Vivec Cantons that Divine Intervention will take you to the garrison at Pelagiad, not Ebonheart.
You'll need a middling (teens or 20s) skill in Mysticism before your chance of spell failure drops to "reasonable" levels with Divine Intervention; reasonable being somewhere in the teens. Not to panic, though. "Divine Intervention" amulets are not too uncommon as loot drops starting at level 4 or so and it seems that almost everyone is giving you "Divine Intervention" scrolls as early quest rewards ("don't leave home without it" seems to be the general sentiment).
Tunifus doesn't particularly care where you get your ingredients from. The benfit of following his suggestions is in the additional rewards (usually in the form of skillbooks or training) rather than the quest rewards themselves. Note that Tunifus' quests are not handled through normal dialog. They are handled as part of his greetings, which makes them work a little differently than the quests for other factions. Also take note that he will give you three pieces of alchemical apparatus as rewards, but that will not include a Calcinator. You'll need to come up with one of those on your own.
Tunifus would like you to get five portions of Marshmerrow for Restore Health potions (he'll tell you the recipe if you ask about it).
He suggests that you take a Fortify Strength potion to a local farmer, Balur Salvu, who lets the Imperial Cult gather from his farm at no charge. The directions to Salvu are pretty good. Go north on the road from Vivec's Foreign Quarter Canton. At the T intersection, turn right (don't cross the bridge to Seyda Neen), and follow the road to Salvu's farm. If you get to another bridge, you went too far.
You do not need to talk to Salvu to complete the quest, but if you do talk to him, you can give him the Fortify Strength potion. In return for this, follow the dialog options and he will give you a copy of "The Cake and the Diamond", an Alchemy skillbook. Otherwise, just harvest Marshmerrow from his fields. You only need five portions to complete the quest, but you can harvest all of it and anything else growing in his fields (ingredients take about 4 months to grow back).
Return to Tunifus to complete the quest and receive a Restore Health potion and an Apprentice's Mortar and Pestle. If you did not deliver the Fortify Strength potion, you can keep it, but you won't get the skillbook. I'd recommend keeping it if you have already read "The Cake and the Diamond" (there are three copies just laying around and it shows up as a random skillbook in loot drops).
Tunifus needs five portions of Muck for Cure Common Disease potions. He suggests going to Gnisis since there is a muck farmer there who lets the Imperial Cult gather ingredients at no charge. While you're there, he wants you to deliver four Restore Health potions to Chaplain Ogrul at Fort Darius. Except for bringing back the Muck, the rest is completely optional.
The farmer, Abishpulu Shand, isn't anywhere around, so you can just harvest from his eight large Mucksponge plants (the ones that look like they have holes in the top). The little plants do not open and look like their tops are closed. You have an 80% chance of getting a Muck from each of them, so you should be able to collect at least five. If your luck is bad, you might need to stop by an alchemist's shop somewhere and spring a gold or two to make up any deficiency (Nalcarya of White Haven in Balmora always has 10 portions).
Chaplain Ogrul's dialog is handled as part of his greeting. If you have four Bargain Restore Health potions in your inventory while this quest is active, he'll just take them. You will not have the option of keeping them unless you choose not to speak to him at all or do not have them in your inventory when you do. If he gets the potions, however, he will offer to teach you one of four "Restore Attribute" spells (Strength, Willpower, Speed or Luck). If you already have those spells, you're just kind of out of luck as far as getting something from him, but you do have a "No spell for me, thanks" dialog option to cover that possibility.
Return to Tunifus with your Muck portions to receive a Cure Common Disease potion, an Apprentice's retort and a copy of 'The Alchemist's Formulary" (recipes for Restore Health, Restore Fatigue, Cure Common Disease, Restore Magicka, Feather, Levitation, Water Breathing, Fire Shield, Fortify Strength, and Invisibility).
Tunifus wants five portions of Willow Anther for Cure Paralyzation potions (he'll tell you the recipe if you ask). Just as an aside, how are you supposed to drink a Cure Paralyzation potion when you're paralyzed? Anyway, he suggests heading to the house of Gurak gro-Bagrat, a local smuggler since he has several plants growing in front of his house. The directions are pretty clear, so getting lost shouldn't be an issue.
gro-Bagrat has seven plants in front of his house, each with an 85% chance of producing the ingredient. If that isn't enough, there are several Willow flowers along the road on the way back to Ebonheart, so getting the full five should not be a problem.
Return to Tunifus with the Willow Anther to receive a Cure Paralyzation potion and "The Four Suitors of Benitah", a Restoration skillbook.
Tunifus wants five portions of Scrib Jelly for Cure Poison and Cure Blight Disease potions. Unlike your earlier retrieval missions, Scribs only have a 60% chance of dropping the ingredient, so this might take a little longer. Also, unlike plants, Scribs may not have many health points, but they can fight back and they can paralyze you (that Cure Paralysis potion might come in handy).
Tunifus suggests going to the Shalk Eggmine (it's just downriver from Balmora), which is probably not a bad idea. There is one preplaced Scrib in the area and are a couple of random critters that might be Scribs, depending on your luck and level. There are two eggminers outside the mine. Dinok (the Redguard) will give you a portion of Scrib Jelly if you talk to him about it. That leaves only three or four more portions to collect.
There are seven preplaced Scribs inside the mine (none in the Queen's Chamber, although they could spawn as random creatures). The Scribs are not hostile until you attack them, but Kwama Foragers are. If you stay out of the Queen's Chamber, you won't encounter any Kawma Warriors. In the Queen's Chamber, you can encounter up to three Kwama Warriors if you're level 3 or higher.
Even with that 60% drop rate, those seven preplaced Scribs should be enough to make your five portions. If not, you'll need to either go hunting scribs somewhere else or swing by an Alchemist to purchase what you need. The base cost is 10 gold per portion.
Once you have your portions, report back to Tunifus, who will give you one potion each of Cure Poison and Cure Blight Disease and an Apprentice's Alembic.
Now Tunifus wants five portions of Corkbulb Root for Restore Health and Cure Common Disease potions (you should already have those recipes, but you can ask about them if you don't).
If you have been gathering ingredients as you go, you might already have your five portions. The Ascadian Isles have lots of Corkbulb plants along the roads as you have run Tunifus' other errands. If not, Tunifus suggests taking advantage of the Imperial Cult's no-charge-harvesting agreement with the plantation owners in the area, especially Arvel Plantation. To help you get there, he teaches you "Swimmer's Blessing", a 5-point "Swift Swim" for 30 seconds.
Corkbulb plants have an 80% chance of containing the ingredient, so you should not have any problems aside from random encounters along the roads as you hunt for them.
Report back to Tunifus when you have your five portions and he will give you one each of potions of Cure Common Disease and Standard Restore Health potions.
Tunifus wants five portions of Rat Meat for Cure Poison potions. He suggests going to the Underworks of any of Vivec's cantons, but any old Rat Meat will do and you've probably run across at least that many Rats in your travels for the other ingredients. If not, head for the sewers.
I'd recommend staying away from the St. Olms, St. Delyn's and Foreign Quarter Underworks because of other nastier NPC encounters in those areas, but just about any other underworks will do. There are three Rats in the Storage Area of the Arena Canton. There are a couple of Telvanni Sewer Rats in the Telvanni Enchanters store (Waistworks) and each can drop two Rat Meats. There are no other pre-placed Rats in the Vivec area, but there are plenty of random encounter possibilities around.
If you don't feel like hunting for it, Chaplain Ogrul is one of the merchants with restocking Rat Meat. Its base value is 1 gold.
Bring your five portions back to Tunifus for a standard potion of Cure Poison and a copy of "The Wraith's Wedding Dowry" (an Unarmored skillbook).
A little easier, but a little harder this time. Tunifus needs some Netch Leather, but only one portion. He suggests heading back up to Gurak gro-Bagrat's house and also suggests that you stick with the Bull Netch rather than the Betty Netch, who can be a bit nastier. Not a bad suggestion since a Bull Netch has four 60% chances to drop up to four portions while a Betty Netch, which is harder to kill, has only three 60% chances to drop up to three.
I prefer to use a "Bound Longbow" since it gets me an extra 10% to my Marksman skill and can do about twice the damage of the more common Longbow and Steel Longbows. Most of the time a Bull Netch is a one-shot kill with the Bound Longbow and Marksman only in the 30s. A Betty Netch is usually a three to four-shot kill under the same conditions.
Take your Netch Leather back to Tunifus for five Quality Magicka Resistance potions, a "Belt of Balyna's Soothing Balm" and +1 to your overall reputation.
Tunifus has no more quests and suggests that you talk to the Almoners or Shrine Sergeants for more work.