ANY race can be best for anything. As the only thing with the races in Skyrim is now about a 10 point increase in whatever skill at best, you can choose whichever race you like the appearence of and if you do activities which may increase a skill, that skill will go up just as fast as it would with ANY OTHER race. The basic bonus of 10 to smithing for an Orc can just as easily be gained by creating armour and weapons at a forge with a Dark Elf character, and the same with the other skills for the other races. The major factor in levelling up is the Stone you choose, which decides how your characters skills are attributed. There are the three Guardian stones close to where you set off in the game and they are the three skills you mentioned in the question: Mage Warrior and Thief. Those three increase the rate at which your skills increase (20% increase) , although you can only choose one at a time. There are other Stones which you can find which give you different bonuses like invisibility to be used once a day, increased carrying capacity and no armour running affect, or various other effects you'll find once you play the game.
If you dont want to hack people up or take damage, then Mage would be your choice. With this choice you focus on your Magicka stat to cast offensive and defensive spells, and leave your health and stamina stat alone. Its cool because you can summon the recently killed bodies of enemies to fight for you, destroy a whole group of enemies in one fire blast (enemies that would have taken a while to kill one-on-one with warrior or thief types), transmute iron ore to gold, become invisible and a hundred other different spell effects.
With the warrior its all combat; attack, block, attack, block, power attack, power block, attack, decapitate flame atronach, repeat. The main thing with the warrior class is that its consistent. You build your armour and weapons skills up as much as you can, and die as few times as possible. Its a powerhouse class that, if used properly, should see you dying a lot less than the other classes, which are a lot more risky. You increase your health the most, then stamina and completely ignore magicka or use it minimally to increase health and stamina as much as possible.
The Thief class is one that doesnt rely on open combat so much, and use sneaking as a standard rule. If you successfully hidden through sneaking, you can do as much as 15 times the damage to an enemy if you sneak up behind them and get a hit in with a dagger, which will kill even the baddest dragon priest. You can use a bow to do triple damage to a far-away enemy, and use rapid follow up shots to finish off his buddies or other enemies if they get too close. You get used to sneaking everywhere, stealing everything that isnt Key Required, and having a massive surplus of stolen goods at your house. With 100 on the sneak skill, youre basically invisible anywhere when in sneak mode, unless someones standing next to you with a torch!
If youre just starting off, dont try to lean massively towards either of the three types, and just enjoy yourself. You can get serious about it later when youre addicted.
And with the races, just pick the one you like the look of the best and maybe try and get him to look like you if you can (although you should be wearing an enchanted helmet most of the time to improve your skills) , there are some bonuses with the different ones but theyre just some powers or skills you can get in the game somewhere and arent unique, just a better start off to some skills to keep the races defferent other than by appearence. I picked the Khajiit because, well, its a cat-person! And you can see in the dark whenever you want, but thats just a cat-person bonus. Dont think too much and just enjoy the game...for the next year or so...