I have a Trading biased city so: Barracks, Training Ground, Merc Camp and only 2 Armouries. There are only 2 Ancient Wonders giving a Fighting benefit, Dragon Abbey (Mages) and Simia Sapiens (units production bonus) and to be fair these AW's are one of the few to exist in all 3 cities. To say this city is WEAK at fighting is a serious understatment.
However, as MinervaOz stated, the key is to understand how effective your different Troop types are Vs the enemy. For example a Light Range Troop choice from each of the 3 sources (Barracks, Tr. Ground & Merc Camp) will fair diffently against each of it's enemy counterparts and against the other Troop choices it has to face. But you have to learn/understand the mechanics of the Fighting system to spot the best tactics. Quite often those squishy Cerberuses, Mages or Rangers are going to get royally hammered BUT knowing the enemy will focus on destroying these units can give your other units all the time they need to kick butt.
My fighting is Auto only. Plenty of Tournament, Spire and Provinces have taught me the ropes. Armed with this knowledge I can honestly say that sometimes this city surprises even me!