I probably worded that part about the barbarians not too well... what I meant was that I got two attacks from barbarians with 5+ attack power (without promotions, I'm quite familiar with bears that have +60% strength, although that is an entirely different scenario which is all fine with me) during my first 5 - 10 turns in a game yesterday. The same happened various times in other maps with similar settings and upon opening worldbuilder, I saw a good deal more giants and thelike all over the map, who are quite capable of killing off any civ before they can even build a second warrior. Not to talk of incoming orcs from cities and other barbarians which will be spawned from lairs if I ever manage to get past those first turns.
I like to struggle and have a hard time defending against incoming forces, but that stops being fun when the map spawns enough barbarians to kill off every civ in a very short time frame right from the start. That isn't exactly survivable when all you have is a scout, warrior and a city to protect.
Dungeons, giant forts and the goblin lairs generally don't spawn mobs for a few turns. So, if you see one, especially a giant fort, try to get rid of it as soon as possible. In my experience, all lairs will pop one mob that will remain stationary until another pops several turns later. The exception to that is a minotaur lair, which will pop a minotaur that will stay inside a two square radius (even if it overlaps your cultural boundaries) until another one pops. Then, generally they hang out together unless you have a settlement near or one of them gets drawn out.
For giants, they can be troublesome. But, I've never seen one on a very early turn where I didn't have at least two warriors defending a town. With the warrior's city defense and +vs barbs (IIRC) bonus, two can generally take down one early game giant. With three, or two and an archer, they're not a problem at all.
Named barbs and things like the nasty ironskin troll/ogre (or whatever they're called) have predictable AI's. Orcus is relatively easy if you can catch him in bad terrain with a few warriors. Otherwise, get six-seven or so and he should go down. The goblin archer named spawns with several additional units. He's not so bad by himself because all he does is use a ranged attack. I've never seen him actually attack anything using a melee attack. So, if you can kill off his adds, let him plink away until you have sufficient forces to counter-attack. ( Just be advised, you can leach exp off of him for your ranged attack units. But, he gets exp for surviving their attacks as well...) For the big troll/ogre guys that can pop on a "bad bad" dungeon spawn prompted by an AI explorer, they don't melee attack either. They will pillage, but they'll only do their ranged attack, so no worries about one eating up your new cities.. Though, they can be troublesome pillagers and weaken units for other barbs to eat.
I've been surrounded by barbs as well. Usually, if it's the kind that will directly melee attack a city, there's not much of a problem unless it's a horde accompanied by ranged attackers or a big unit. But, they can be a problem in hampering your expansion and development. The last game I had where that was a problem, I had barbs on every side of my small, three city empire, including Minotaurs, the named gobbie archer, a couple of giants and random goblins. I survived by pumping out warriors and rushing to archery. The key is knowing what their AI is likely to make them do.
Doublecheck to be sure you're not using a custom start, like rampaging barbs or something. There may also be some bonuses due to barbs depending on your difficulty level. FFH2 turns barbarians and animals into a significant game element. But, all but the strongest of them become non-issues fairly early in the game.