I've played all the races. I was able to crate a pretty fun Argonian character using porchdrinker's random character guide (see the tutorials forums). He's a social climber, and wants to gain status, because he's terribly envious. I've forgone the main quest entirely--instead of going to Riverwood, he wandered around, raiding bandits for money and collecting ingredients to make potions to sell (he spent a few ays at teh Alchemist's Shack), before he went to Riften. Once he got to Riften he sold his poin for some coin, made freinds, bought a house, got a thaneship and went to Windhelm to schmooze with the Argnoan dockworkers and oter Windhlem folk and to court and eventually marry Shahvee. Now he struts aroun Riften in his fancy clothes like a lord and is all "How you like me now?" to Talen-Jei, his rivel (at least in his own mind). Now he need to do something to support his lavish lifestyle and to gain more clout so he's going to join the Theives Guild.
It is a little interesting that with the Argonian or Khajiit, you reallly feel more like an outsider, which is the complete opposite from what it felt like for me to play a Nord. All the other races are kind of somewhere inbetween--the Imperials and Breton fit in a little more easily than the redguard, Bosmer or Dunmer. I've only played one Altmer and although he didn't feels like an outsider it seemed like there's not a lot in Skyrim for an Altmer to do, I mean, besides walk around in Thalmor robes and sneer at locals. If I ever create another Altmer character, it'll have to be an exile or a rogue of some kind, more of rebellious fringe type.
Some fun thing (for me at least):
--create a personality, likes and dislike for your character and develop habits around them. My Argonian for example hates cooked fish and will only eat raw fish (his favorite is apple cabbage soup with a piece of raw salmon). He also like to take a swim in the lake by Riften after nightfall. And of course, when he's in Riften he always wear fine clothes instead of his armor--it helps to remind people he's a Very Important Argonian.
--create motivations for your charcter and focus on them, rather than doing everything that comes your way. Also have a different starting location othe than Riverwood or Whitern. With my Argonian, he hsn't gone to Riverwood yet. Instead he went to Riften because his backstory is that he did something pretty shameful and had to leave Riften before, and nwo he's returning to redeem himself. He's limiting himself to just the eastern part of Syrim, at least until he joins the Theives Guild. I've had character who hung out in Falkreath, Rorikstead, Riften, Ivarstead or Solitude before heading to Riverwood or Whiterun for the first time.
--I've done a couple "no dragon" play-throughs where my character had no interest in informing the Jarl about the Dragon at Helgen, and when I did Bleak Falls barrow to befriend Lucan, I left the Dragonstone on the boss draugr. I did that with my last Bosmer--she only has gone to Whiterun once, to complete the Night to Remember daedric quest and instead of asking to see the Jarl at the city gates, she bribed her way in to talk to ysolda. ther than that, she has had nothing t do with Whiterun at all. It is quite a differnet feel to wander around Skyrim without any dragons to worry about, or to never go to Whiterun. (Note that without taking the Dragonstone to Farengar and completing Dragon Rising, you will not be able to complete the Civil Wat QL as the Main QL and the Civil War QLs are linked. Also the shout ability cannot be unlocked without finishing Dragon Rising, ever though you can still collect Words of power from the Word Walls--you just can't use them. So only try a no-dragon play-through if you dont want to do Civil War QL and you can do without shouts.)
--One play-through that I really enjoyed and want to try again some day: I had a Breton battlemage who went around collecting shouts from Word Walls around Skyrim, via teh "letter from a friend", where you use a shout in city or another location, and you get a letter directing you to a new Word Wall. Then you go find that Word Wall, learn the Word of Power and then repeat the process. For the early part of her game, that was her main objective. I was able to get teh Thu'um master accomplishment with that character that way (you do need to the main quest to get that accomplishement though, as 2 of the shouts can only be learned from the Greybeards and a third from entering the Time Wound on the Throat of the World)