I am about to ding 33 and have not set foot in Diamond City yet. I have been carving a path of naked headless corpses from north to south and picking up side quests along the way. I also craft compulsively, even shit I don't intend to ever use, just for the XP and resale value of the items. Really, simply sweeping around Sanctuary and the fort town just to the SW of it will get you set up pretty well and close to L15.
But to add to what hodj said for Noodlebro, here are some basic tips I picked up watching my wife play, who also struggled some at first:
Companions:
Wadsworth is really good early against anything not a Mirelurk. In particular, the Forged guys get rolled by him because the flamers they pack do not harm him very much. Even if you are planning on going lone wanderer, use him for a while until either your stealth or combat skills are up to snuff.
Shotgun:
Double barrel will get you by early, but the two shot limit makes it worthless in the long term. Get a combat shotgun (several vendors sell them) and mod the shit out of it. Go full on maximum damage. When a fight goes south, you will be able to pull a win out of your ass more often than not by just whipping out the CSG and firing indiscriminately.
VATS:
Even if you are not building around it, learn when and how to use it for close quarters fighting. Hodj and Vorph posted some good info on it, but I would add that using it to locate enemies by tapping in and out of VATS is another way to get through the early parts of the game, especially if your Perception is kind of shitty.
Damage Types:
Be aware that some enemies resist certain damage types more strongly than others. Pack at least one ballistic and one energy weapon at all times, because sometimes that guy who laughs off your pistol shots might die in one or two hits from a descent Institute or Laser weapon. Eventually, once you get some combat damage ranks in your preferred weapon, you will simply overpower defenses. But initially, covering your bases is important and also keeps you from being overly dependent on a single type of ammo.
Endurance:
This stat makes an enormous difference in the amount of HP you have. You don't _need_ to have a high base END, but if you skimped in this stat, consider wearing END boosting items (Sea Captains Hat, some cloth outfits) to add 2-3 points as this can boost your HP total a good 20-40%. Also, always take every available rank of Toughness (the END 1 Perk) because it is the easiest source of defense there is.
MedX:
If you abuse no other drug in the game, you should consider using MedX for tough fights. The extra damage reduction can make the difference between getting gibbed or being able to take a few hits, especially very early in the game. Yeah Psycho is an early I win button too, but MedX is plentiful and has a low addiction risk.
