Oblivion Mod Manager [http://www.tesnexus.com/downloads/file.php?id=2097] - Helps you sort out mods, and is much better then the standard one the launcher uses.
Oblivion Script Extender [http://obse.silverlock.org/] - Greatly expands what Oblivion is capable of. Required for more of the interesting mods.
Unofficial Oblivion Patch [http://www.tesnexus.com/downloads/file.php?id=5296] - Fixes so many small bugs in Oblivion its not even funny. I'd say its the only "required" one if you want an enjoyable experience.
Also if you got Shivering Isles, I recommend the Unofficial Shivering Isles Patch [http://www.tesnexus.com/downloads/file.php?id=10739]. Does what the normal one does, only in Shivering Isles. I also recommend the Unofficial Official mods patch [http://www.tesnexus.com/downloads/file.php?id=9969] that fixes issues with the official DLC. That is if you have any.
Natural Environments [http://www.tesnexus.com/downloads/file.php?id=2536] - Adds much more to the ambiance of the world.
Midas Magic [http://www.tesnexus.com/downloads/file.php?id=9562] - Makes magic so much more fun. Adds lots of awesome spells. Including the ability to summon an Atronarch made of watermelons.
Deadly Reflex [http://www.tesnexus.com/downloads/file.php?id=8273] - Makes Combat much more intense. Requires the Oblivion Script Extender. And there's a saying among the community. Once you go Deadly Reflex, you can't go back.
Unique Landscapes Compilation [http://www.tesnexus.com/downloads/file.php?id=19370] - Greatly diversifies the landscape of Oblivion. This is a compilation of all the separate mods that change things, making it one easy install using the Oblivion Mod Manager.
Oscuro's Oblivion Overhaul [http://www.tesnexus.com/downloads/file.php?id=15256] - Overhauls the game adding shit-tons of new monsters, quests, NPC's, fixes for various things(including the leveling system) and much, much more. I can't play Oblivion without 3 mods. This is one of them. The Unofficial patch is the other, and BTmod is the other(more on this later).
Reneer's Guard Overhaul [http://www.tesnexus.com/downloads/file.php?id=5977] - Overhauls the guards. Makes them less psychic and makes the guards much less, well annoying.
ImpeREAL City [http://www.tesnexus.com/downloads/file.php?id=19589] - Makes the districts of the Imperial City more unique and interesting.
BTMod [http://www.tesnexus.com/downloads/file.php?id=1825] - Greatly improves the inventory system. This is the third mod that I consider "required". It really, really helps when you have a lot of stuff in your inventory. See the screenshot on the downloads page(click images, then user uploaded images).
LowPoly Grass [http://www.tesnexus.com/downloads/file.php?id=5434] - Reduces the polygon count of grass. Resulting in much, much better performance. And no, there is not a noticeable visual difference.
And that concludes the mods I feel are needed for a greatly improved experience, now for some of my favorite mods that add new things like quests and stuff.
Loading Screen Quotes [http://www.tesnexus.com/downloads/file.php?id=5890] - Adds quotes from famous people to the loading screens. It replaces the boring tips that normally appear.
Better Letters [http://www.tesnexus.com/downloads/file.php?id=5392] - Replaces the generic letters and notes you find everywhere with more interesting and funny ones.
Akatosh Mount [http://www.tesnexus.com/downloads/file.php?id=10819] - Adds a ridable Dragon. 'Nuff said.
The Lost Spires [http://www.tesnexus.com/downloads/file.php?id=12997] - This mod is so good, it got the creator hired by a game company.
Lost Paladin of the Divines [http://www.tesnexus.com/downloads/file.php?id=10505] - Adds a quest for a new set of armor for "good-guy" characters. Plus I helped test it, so yeah.
The Pirate Isles [http://www.tesnexus.com/downloads/file.php?id=28290] - Great for any Pirate enthusiast. Lets you become a pirate captain, build your crew, and attack ships at sea.
Colored Map [http://www.tesnexus.com/downloads/file.php?id=2308] - Color's the map. Making it more aesthetically pleasing to the eye.
Elsweyr Anequina [http://www.tesnexus.com/downloads/file.php?id=25023] - Adds the province of Elsweyr to the game complete with new cities, quests, landscapes, ect. You'll have to download from the mirror sites though(click "mirrors" then the link). I also suggest perusing the add-on mods for it which are in the description page. Also, if you decide to use the colored map above, I greatly recommend the Colored Elsweyr [http://www.tesnexus.com/downloads/file.php?id=25150] add-on.
And that should be good for now. I suggest browsing the Top 100 [http://www.tesnexus.com/downloads/top/alltime.php?adult=2] list for anything that catches your fancy(should probably set it for "non-adult" if you want to avoid the more... "sexy" stuff). I also suggest just looking through the Nexus in general. Choose a category and just browse. There are plenty of other mods I haven't mentioned that are just awesome. Like Malevolent [http://www.tesnexus.com/downloads/file.php?id=17781], which adds a sentient suit of armor and a multi-branching quest-line.
