I think your best bet is to post a game.
Some basics.
Spearmen /pikemen get 100% defence against horseback. Best to use melee against these. If a city has a mix of units you need to be careful.
Longbowmen are good city defenders. Always use catapults to bombard to reduce any city defence bonuses to 20% or lower and then weaken by attacking with catapults to cause collateral damage before sending in your melee units. Catapults are throw away units in my game as you normally save on melee units who gain greater experience. I always like at least a 2-1 majority compared to city defenders if i dont have strong enough troops. Macemen would be more ideal against longbowmen backed up by catapults to soften up their blow.
Logic: If you attack with 3 units against 2 defenders and dont take the city the units are lost in vain as you can bet a new unit will appear the next turn.
Never attack across rivers.
Axemen are good against melee units like swordsmen and attacking archers and catapults/ spearmen in cities. You really need to look at what defence bonuses units get against the others. Its in the manual!!!
Beware cities on hills that get +25% defence from hill. Useful to build units from cities with barracks so they stand a chance when attacking. Beware cultural bonuses.
Attacking Russians who have cossacks is very brave move.
I think numbers is probably your problem. Target your attacks on one city using numbers. Attacking 4-5 cities at once is suicide unless you have swarms of troops. I always have at least 2-3 cities out of 10+ cities producing units 80% of the game to stay in touch with the AI till 1300ad. Never attack someone unless you think you can win. If your average city has 1 defender you are way under strength. I like 2-3 defenders per city plus an attack force of 10+ units to swell my border cities to 6-7 units and to reduce and cripple my smaller neighbour.
As a starting strategy try building 3-5 cities quickly, (Dont worry if science drops to 50 or 60% it will recover) developing bronze working asap linking up copper and attacking with 5 or so axemen against a city with 2 archers before 1000bc-500bc on standard speed. You will soon learn the ways of war.
The new nerfing of chopping forest may slow this strategy down but the axemen rush is still a great starting move as long as mansa and his skirmishers are not about and your enemy does not have copper/iron linked up.