Dealing with barbarians early in the game

The official Civilization Youtube channel actually has a video on this. The key take away here is that Barbarians now send out scouts to seek out civilizations. Once the scout finds a civilization it runs back to it's Barbarian Encampment, which will then start to spawn military units to attack the discovered civilization. Until the scout reports back the Barbarian Encampment won't spawn any military units, with the exception of one garrisoned unit (which in my experience was always a spearmen).

So, with that said, the best way to deal with Barbarians is to either never let the scouts report back to their encampment to begin with, or to just follow them back to their base so you can wipe out the encampment before it has a chance to spawn too many units. Since scouts are pretty fast it's likely that you'll be doing more of the latter than the former, but if you keep a few military units (including Archers) near your borders than you might be able to wipe out the scouts before they report back. You'll still want to eventually track down the encampment and wipe it out, though.

Army wise you shouldn't need anymore than four units (two ranged, two melee) to deal with any encampment you come across, but should be able to just get away with two units if you make sure to hunt any encampment down the moment a scout appears.

Finally, and this final part is just speculation for now, but I think a scout only counts as discovered your civilization if it comes within line of sight of your borders. If true, then you could just keep a few military units stationed outside of your borders to scare off any scouts before they see your borders. If they don't discover you then they'll never retreat back to their encampment to report your existence in the first place, so you don't need to worry. Again, though, this is just speculation for now.

Yes I found this to be the case as well. they have had a significant boost to their spawn times and aggression. However they are quite comfortable with leaving their encampments now instead of fortifying in them so they can be easily pulled out and taken with 2 units. The best way to take them down is to of course use the environment (hills/rivers etc) and research Archer ASAP. The slinger is great in the way that it can attack without retaliation, however they have terrible damage. Archers make short work barbs.

I have played the games on almost all difficulties now and I have found a reasonably cheap and viable strategy for most cases. This is based on you scouting your vicinity early, simply take your starter unit warrior and walk in a semiwide circle around your first city. I always start with a builder and then another warrior or, on lower difficulties a scout. By the estimated time your second "combat" unit is ready your should have mapped out the surrounding area and probably found a barbarian camp already. Your warrior should be just so strong that he can take out the spearman guarding the camp on his own. Worst case scenario he will need to rest one turn, but then again, he will gain a level from killing the spear so he should end up on minimal 61 hp after you promoted him(50 heal from promotion + 1 turn inactivity that gives 10 health).

Now, any new barbarian camps in "the explored" area will be visible when they spawn and announced by a small note. As soon as it does, send one of your warriors there to wipe it out(this will be easier with the +5 attack bonus from the early policy) and let your other warrior heal up close to your base. Rinse and repeat until you are ready to settle your second city. Once you have build your first settler, build another warrior(they are the cheapest and you don't need the spears for barbarian camps) and let him follow your settler when complete and let that warrior guard your new city.

If done correctly you will have 3 warriors and two cities. One will scout out the immediate areas while the other two go camphunting(this is their ONLY job). IGNORE the barbarian scouts! They will not pillage, they will not steal your workers or settlers and is only a minor nuisance sometimes when they get in the way. Killing them off costs you around 40hp each though and will make it so you can't kill of a freshly spawned camp if playing on higher difficulties, and make no mistake, this will end you.. Once you have settled your third or fourth city, consider building ancient walls in the ones that you plan on using as bottlenecks for enemy attacks(if you can) although I usually prefer having a bit larger army for that amount of upkeep unless I am close to an AI.

Hope this was helpful and good luck!