I have played with quite a bit in most races in Skyrim and only a select few in Oblivion.
Argonian - Great advantages when it comes to mostly disease resistant (can still be poisoned but much more rare), water breather (can travel via safest routes in Skyrim, you can treasure hunt sunken ships and can discover many landmarks thanks to the Argonian alone), skills for Assassin/Thief built in, look cool, sound cool, can now wear and use everything everyone else uses and wears. The funny thing is, my Argonian character "Rumar" as you can see in my avatar and signature, has done it all, he is the College Mage at Winterhold, he is the Listener and the leader of the Dark Brotherhood, he is the new leader of the Thieves Guild, and he is the new Companion Leader. All with just the Argonian which is just awesome!
Khajit - great as well because similar skill build up as Argonian, they have night vision (which can also be gained with Vampirism for other races), most of the time I find myself out at night even though I should give my character rest so I'm still getting used to making them sleep instead of downing health potions and recovery spells. Their Histskin recovers your health 10x quicker than normal for a minute which can
really help out when you're outnumbered or outmuscled. They look and sound cool as well (favorite voice in the game are the male Khajit voices by Kharjo and J'zargo), they are also very fun to use as well as the rest IMO.
Imperial - Great way to start the game when it comes to money, they have the built in luck factor and can find more gold and jewelry in containers and looting from bodies than the rest, in dungeons otherwise pretty dry when it comes down to it like Bleakfalls Barrow, my Imperial has found numerous of jewelry as well as a decent amount of gold. Their Voice of the Emperor is very helpful too when it comes down to taking out your targets or defending yourself against bandits and others who decide to jump you. Calming them regardless of level (except dragons), you can get an easy kill and the others are still under your voice effect so even if you kill their leader such as Kazim from "In My Time of Need" quest in front of his men, you can still walk out no problem. In literally half the time, I was able to buy Breezehome in Whiterun and the others in other cities than any other race. So if you're worried about gold and having your inventory balanced, I suggest going Imperial.
Orsimer/Orc - Tough, built in smithing bonus as well as Heavy Armor and Two handed which is fun to use even though I prefer my one handed + shield. Their Berserker Rage also comes in handy when in a tight situation just like with the Argonian's Histskin ability, you take half the damage and deal double the damage. If you're outnumbered, up your one handed skill for the double wielding flurry and just devastate the field/dungeon. Not much else to say other than if you wish to be a tank, go Orsimer.
High Elf/Altmer - Great when it comes down to the magic users, I'm not so thrilled with strict magic based which tends to be a bit harder than just combat with armor and the rest to help protect you from serious harm as opposed to 0 defense with robes. If you're patient enough, magic based users can definitely become great and almost unstoppable. Be warned for Bretons and Nords have magic resistance which help protect them a bit moreso than other races.
Special abilities
Highborn: Regenerate magicka faster for 60 seconds, once a day.
Fortify Magicka: Passive increase in magicka by 50 points. If the Necromage perk is taken from the Restoration perk tree, this will increase by 25% to 62 points.
Skill bonuses
+10 Illusion (level 25)
+5 Alteration (level 20)
+5 Conjuration (level 20)
+5 Destruction (level 20
+5 Enchanting (level 20)
+5 Restoration (level 20)
Starting Spells
-Flames
-Healing
-Fury
Dark Elf/Dunmer - Fire resistance and well known to be the better race to use when you're a vampire since that built in fire resistance helps against the vampire's weakness. The Ancestor's Wrath can help wipe out a group of weaker enemies as well as a group of bandits or Draugr that come after you which if you have fire enchanted weaponry, you're set with this. It's a decent boost in the below bonuses and another great choice for mages.
Skill bonuses
+10 Destruction
+5 Alchemy
+5 Alteration
+5 Illusion
+5 Light Armor
+5 Sneak
Starting spells
Flames (Destruction)
Sparks (Destruction)
Healing (Restoration)
Special abilities
Ancestor's Wrath: Once per day for 60 seconds, opponents that get too close take 8 points per second of fire damage. Ancestor's Wrath can harm neutral NPCs such as town guards.
Resist Fire: Your Dunmer blood gives you 50% resistance to fire.
I plan on starting on a Redguard for my girl and Bosmer/Wood Elf for myself in Skyrim, I will give a more detailed review on the race (another great assassin/thief) when I begin on that. I hope to start recording though with the Video Capture device which I still lack the money to get.
Maybe a serial rapist imperial?
Boy, wouldn't that spice up gameplay?