A game of Explorers & Exploiters typically has three phases: the Explorer Phase, the Barbarian Phase and the End Game Phase. In the Explorer Phase, the players strive to expand settlements (and place the big settlements), in the Barbarian Phase to expel settlements (and place small settlements) and in the End Game Phase to place the few remaining settlements. However, they're not clear-cut - you may very well have to throw in an early barbarian action or a late explorer action if the current game state calls for it. Let's look at them one by one.
You start with an unknown game board and the only way to find out more about it is to explore it. Choose any explorer and move it around to explore tiles. Do NOT exploit the tiles until you're ready to connect foundations and resources. Not only would this cost you an action but you would also give away your knowledge monopoly. Until a tile has been exploited (turned), only you know what lies beneath it.
Should you then explore on your own or follow another player closely? Given that no other player may use "your" explorer until your turn has passed again, you can be sure that no other player may capitalize on your exploring. If your explorer exploits a foundation, you will still be able to use it to exploit resources in later turns until you decide yourself to give up the explorer and use another one. Given that two explorers start at each starting tile, you could use the second explorer to follow the first explorer. However, this strategy is only successful if you immediately after the first player's exploitation can exploit a connected tile and expand a settlement. Generally, I recommend that you explore an area on your own and abandons it only if other players seems to have found better areas.
What is a "better area" then? The answer is an area with a high ratio of resources to settlements. Why so? The probability of finding new valuable tiles is independent of previously found tiles in the area. Yes, but a high ratio means bigger settlements and your objective in the first phase is to place your big settlements. A city needs to be connected to all four resources and the sooner you can get rid of your city, the better. Once your big settlements are gone, you may enter the Barbarian Phase.
The barbarians have two values in the game. First and most important, they may derive settlements of resources so that the player may expel them and place smaller settlements instead. Second, they may interfere with other players' intentions to explore and exploit. However, don't make the mistake of believing that expelling settlements harms the player who built it. Once a settlement is placed on the board, the player's objective of getting rid of a settlement is fulfilled, and nothing that happens afterwards will change this. You should only expel a settlement if you can benefit from it by placing one of your settlements. If you don't have the corresponding settlement on hand, don't expel.
But barbarians can not only be used to place smaller settlements. If you disconnect a settlement from a resource, that resource may be used by another tile, so if you play your barbarians well, you may both expel and expand in the same turn!
As stated in the beginning, the Explorer phase and the Barbarian phase may very well be over-lapping. If your explorer haven't found anything useful, it may be better to let a barbarian take advantage of the other players' findings. It's easier to exploit a known tile with a barbarian than an unknown tile with an explorer. If your barbarian have raged through an area, it may be time to throw in an explorer to re-exploit the tiles (but beware so that not another player takes "your" barbarian and attacks you).
End Game Phase
When you have got rid of your biggest settlements, you should ideally only have a couple of hamlets and a village or town left to place. It's now time to search for the best opportunities to place them. The "best opportunities" include tiles that can be reached in as few actions as possible but also tiles where your plans may coincide with other players' plans. The latter is a very important point - don't make the mistake of only counting the number of settlements your opponents have left and only thinking of them as opponents. You need to understand what they need to place and adopt you play accordingly.
Take an example where you need to place a town (3) and your opponent a village (2). If you can predict where the village will be placed, you may be able to connect another tile and place your town. Likewise, if you need to place a village, beware of giving opponents the opportunity to place towns. If you have to give away an opportunity, this is usually OK since it will place both of you in a relatively better position than the other players, but be sure not to give that opponent the victory.
Finally a few words about battles. Should you waste time on battles and if so, when? The first answer is that a battle isn't a waste of time if you're the last player entering it. The reason is that no other player may use the units in the tile until the last (your!) action marker has been removed. This means that you'll also be the first player to use the surviving units in the tile. Provided that you're the last player entering the battle AND the tile is valuable (it may be exploited to connect or disconnect a resource to or from a settlement), you should definitely enter the battle.
A special case is battles around settlements. NEVER EVER leave barbarians alone on settlements. A barbarian on a settlement will be able to exploit one settler each turn and consequently place one settlement each turn. An explorer may, thanks to the support from the settlement, defeat such a barbarian. If the resource tile or tiles from which the settlers were exploited are still available, the city may be expanded again, but even if no such incentive exists, other players should be stopped from taking advantage of undefended settlements.