From someone with almost 400 hours on Fallout 3, here are my tips:
Most important : SAVE ALL THE TIME.
The bugs will get you if you don't.
Most important skills:
Small Guns
Repair
Lockpick
Max these first. Loot and repair everything you find, it will increase the sell value.
Steal everything you can get away with stealing, and sell it.
Once you have 1000 caps, go to Canturbury commons, talk to Uncle Roe about the tradrers and invest in Lucky Harith.
You will now have a trader with an infinite supply of good weapons. Wait in 2-hour gaps to make the different traders spawn.
Go to the Republic of Dave get Ol' Painless. (you can do this peacefully, google it to see how to rig the election). It's a unique hunting rifle which due its no-sway and plentiful ammo, will be your best friend for most of the game.
Talk to everyone. Don't go around randomly killing NPCs, you may miss out on quests.
Take the time to ignore quests and just go wandering the wastes, you'll be surprised how much the game rewards random exploration.