Caveat: It's been a while since I fought without any boost from buildings; I may be overestimating your troops slightly. To compensate, you may want to use both the Tavern boost
and an inventory boost. Preferably, you should need only 2 attacks on an enemy unit to take it out. If you need 3, you'll have to get Zeus up and running.
First, if you're going with only slingers, get to a proper PC, if you're not already at one. You'll need to stay outside of reach as much as possible, which is easier on the Web client. Hover the mouse to see the reach of enemy units. Attack the horsemen first, while keeping your slingers outside the range of the enemy slingers. Boosted, 8 slingers should make short work of 4 horsemen. The slingers won't do any damage on their turn, since you're out of reach. If they're "mean" enough to make it impossible to stay out of reach while attacking the horsemen, try again; they won't always be.
For the second wave, you do the same: take out the horsemen first, staying out of reach of the spearfighters, then take out the spearfighters.
If you can get 4 rogues, that's great. In that case, I recommend going with horsemen of your own. Hold the horsemen back, and move the rogues just within reach of the enemy horsemen, outside the reach of the enemy slingers. The enemy horsemen will attack your rogues, and loose half their health points. The enemy slingers won't be able to do anything on their turn. You may want to preserve your rogues; in that case, move your converted rogues (they'll be horsemen) out of reach of the enemy slingers. Take out the enemy horsemen with your horsemen. Now you'll loose 2 horsemen to the enemy slingers, leaving you with 2 horsemen and 4 converted rogues. Take out 3 enemy slingers. The last will injure one of your horsemen, then you'll take it out.
For the second wave, you let the 2 enemy horsemen and 2 of the enemy spearfighters attack your rogues, staying out of reach of the third enemy spearfighter. You can wipe everyone out with your second turn.
There's a third option, using only stone throwers. You'll definitely want to use both the Tavern boost
and a boost from inventory. This is a retreating strategy. You need to take out the horsemen first, while staying out of reach of the slingers, then take out the slingers. For the second wave, repeat, only now staying out of reach of the spearfighters.
You may also try experimenting with only spearfighters; they're cheap, fast to produce, and should be adequate to the task, if used properly. They're harder to give a blow by blow for, though.
Of course, 7 rogues + one regular is always the best, but probably out of reach for you. Note that everyone has some troops that they find easier to work with. If none of this works for you, keep trying your own ideas. I remember I preferred to work with 4 slingers and 4 warriors (I wouldn't use that today, though), or 8 slingers, depending on the situation.
Good luck, and have fun!
Edit: Replaced one instance of "archer" with "slinger".
