On prince you could probably get to 5-6 cities peacefully in 70 turns if you have decent early production/growth and can bang out early settlers. If you are aiming for peaceful you should build just enough military to handle barbs, deter other civs, and get eurekas - so probably 1-2 slingers (later upgraded to archers) and a second warrior. If you don't get caught that should be enough to work together to clear barb camps and provide defense. Otherwise just build a scout for huts/CS meets/exploration, monuments/builders when needed, and focus on churning out settlers (maybe a gov plaza and ancestral hall to help with that also).
If you want to mix in aggression, that may depend on resources - if you have horses nearby, rushing to horseback riding with a few archers mixed in is often sufficient to grab several cities early. If you have iron nearby, an alternate path (or mixing the two) would be to go straight to iron working and upgrade a couple of early warriors into ngao mbeba (and/or build some outright, since the iron requirement is not too high). They will be slower movement-wise than horseman but can shrug off archers and take out cities pretty readily.
The two biggest general tips (in my opinion) for dealing with early barbs/aggression are (1) get archers asap and (2) keep units together. Once you have archers you should not have barb issues until they get to swordsmen, unless you get caught out in the open alone - which leads to the second point of keeping a ranged unit together with a warrior. But if you play somewhat cautiously, a warrior plus archer should be able to clear out any barb camps you need without much issue.