I went a dark elf because of the natural tolerance for fire, dragons and all that, seemed like a sensible choice. I didn't care for the Nord race, even though the game kinda steers you that way - I thought they all looked too generic. The cats and lizards are just too rediculous for me, they don't look human enough for my liking.
Once you decide your race, play through some early quests, then stop and decide how you want to progress. For instance, there are several side quests - you might have more fun playing those than the main quest. One good skill to go for is blacksmithing, try and get that to 60 as soon as you can, because then you can improve enchanted items, and that's pretty vital. Maybe go hunting for ore, get a pickaxe and a bow, hunt for animals for food and skin, hunt for ore deposits and mines. Then with the leather and iron, make tons of iron daggers to boost your blacksmithing.
I found the bows to be very powerful, more useful than my greatswords, and they offer a couple of benefits - like letting you zoom in when aiming, and even slowing down time when aiming. Very handy stuff if you fancy a stealthy, strategic approach to Skyrim.
I guess my main nugget of advice would be to take your time. Prepare for quests properly, if you don't have a lot of health potions or weapons, then deal with that first - the worst thing to happen is for you to be stuck somewhere during a quest with no easy way out. If things get too much, don't be ashamed of turning the combat difficulty down for that quest.
Thing is, you get no clue about the difficulty of a quest, or even if you've discovered the location already - so don't think that just because you got a side quest, that you can even get there, are experienced enough, and completing the quest is even possible. Some quests need you to do other quests first, that sort of thing, so take your time deciding your next quest, and try and mix them up, give yourself a varied job history before worrying too much about defining your character. Also don't think that you have to spend skill points right away, I would advise saving them, and using them when your skills are enough to get the perks you actually want.