New to FO3

Fallout 3 (F3) is just different enough from Fallout New Vegas (FNV) where some of the advice may not apply,  but I'll give it my best. 

• Keep note of Karma if that means anything to you.  Karma is pretty blanket in F3,  so if you're good you'll be known as good everywhere, even in places you've never been to.  Same goes if you're neutral or evil in Karma. 
• When starting off the Bethesda ruins are a good place to go to.  Raiders tend to re-spawn there about every 72 hours or so, so you can stock up on weapons, armor (theirs) & ammo fairly consistently early on.  Also the Super duper mart has them.   The Wasteland Survival Guide quest is good for both exploring & getting supplies early on.
• Look for the Bobble-heads, there's 20 of them in the game.  They're worth searching for since they give either improvements to skills or stats.   There's one in Vault 101 when you start off, another in Megaton & a 3rd at the Bethesda ruins.  All 3 of these you can & should get fairly early on.  Look especially for those Bobble-heads that "sinc" with the way you wanna make your character.  For example:  I basically made an explorer type so I concentrated on those stats, Skills & perks that would enhance that (Strength, Endurance, Luck; for skills Guns, Repair, Lockpick & Speech).  I also have wanted to have a more "passive income" so I did the Reillys Rangers quest & Lawbringer perk as soon as I could. 
• You get a perk every level, while you don't necessarily have to plan ahead on every little thing, it does help some.   
• Your Repair skill.  Lets face facts you are gonna be repairing just about everything you get,  so TAG that if you've not done so, &/or at the very least spend a few points on that skill every time you get a new level.  
• Things tend to level with you,  so level grinding doesn't do as much good as with other RPGs.  
• Don't kill the merchants that you see roaming around;  once they die that always means that where you live won't get restocked on supplies & caps (money). Same thing if they get killed by something else.  
• Just about everything edible is irradiated,  so if you get the home in Megaton get that doctors area for your home ASAP (removes rads, heals limbs, restores health). Otherwise you're paying a doctor 100 caps to remove those rads,  there's also radaway but that's more random to come by. 
• Save often & have more then one save.   While F3 is pretty stable at this point,  I don't like leaving it to chance.  

That's about all I can think of;  phyrian & MaulYoda, if either of you read this can you add to what I missed?

Frakkin' time expires.  

I forgot about VATS; learn it, love it, use it.  Hit locations are a beautiful thing.   Hit  locations in VATS only occur with ranged weapons though.  

Correction on the merchants:  While they are very useful to keep around, eventually (after having done a quest) becoming some of the best merchants in the wasteland and having the highest merchant repair skill outside of DLC, killing them won't have any effect on any other merchants.  Even if they all die off, the stationary merchants will have updated caps and inventory periodically (every 72 hours, if I remember correctly).  If you want to keep them alive (you do) you can try to reverse pickpocket better weapons and armor into their inventory and they will make use of them.

The Super Duper Mart just east of Megaton is a good place to store surplus equipment.  Once you've killed all the raiders in there, they don't respawn and to my knowlesge the Mart's containers never reset, so you can keep your extra weapons and clothing in there with no risks (unlike the Megaton house).  

I've heard other people mention problems with the Megaton house, but that glitch was supposed to have been fixed with a patch a long time ago, and I personally never had any problems at all there.  That's over 3 of my 4 total characters (one stayed in Tenpenny) over hundreds of hours of game play.  I don't know what conditions may have caused the problem for those that suffered from it.

Oh, that's good to know.  The last thing I want is all my hard earned items being replaced with a coffee mug!  I check wiki and it said the problem was still there?

Hmm, I just checked the wiki and it looks like they may have removed that bug from the list.  In any case, and as always, keep lots of saves.  I say 5 is the bare minimum, I generally like to run with 10 or so per character.  That way, if the worst happens, you can reload without losing too much.

I just started this game, and like many other newbies, I feel really overwhelmed. I'm not sure what I'm supposed to be doing after leaving the Vault. I have the the GotY edition, so I know there's the DLC chapters included. I know one response said head to the Bethesda ruins, but I have no clue where that is. The first thing I happened upon that I've been exploring is the School.

well you can do anything you want the main questline or at least do the Operation Anchorage DLC for the good stuff at the end of the DLC. never heard about a quest that tells you to head to the bethesda ruins, which you can find by following the collapse highway that pass by springville heading in an east direction.

It's not a quest per~se, unless you want to find as many Bobbleheads as possible.  I view it as an early "stock up" place, same with the school.  

@Kimieva:  My thing is exploring, the more places you find the more waypoints you get.  The more waypoints you get the easier it is to fast travel  from point to point later on.  Remember it's up to you how to proceed,  do you want to find dad or make your own way;  do you want to be good, neutral or evil?  What type of characters interest you (Rogue-types, explorers, "techies",  bruisers)?   I don't know if you've played other Western style RPGs (Wrpgs?), particularly from this company but it's pretty easy to get overwhelmed since it's so freeform.