The only time I ever bother with archery at all is if I'm rushing without bronze and all I have is horses, to get horse archers (or If I'm playing as the mongols to get Keshiks). Don't play defensively, sitting around in your cities garrisoned with archers is an open invitation for your enemies to march in and start pillaging your improvements. It's better to have offensive units near the border to 1. Smash any attacks made towards you then 2. Invade the enemy. If you rely too heavily on defensive units you're at the mercy of your enemy. Even if he can't take your cities, you have no way to throw him out of your territory because defensive units suck on the offense(at least until later in the game, especially after riflemen become available).
If you're a beginner here's some tips for taking cities
1. Get a large stack together, mostly consisting of the strongest unit you have at taking cities(give them city raider promotions if at all possible), also include other units, such as axemen or maceman to ward off enemy melee troops and finish off badly wounded garrisons, and some anti cavalry units such as war elephants. Get siege units as soon as they become available.
2. You want to take out large, well garrisoned cities first to cripple your opponents war effort, unless you have a strategic need to capture different cities.
3. Early game expect to lose about 3 warriors for every warrior garrisoned in a city, 3 axeman for every archer and 2 swordsmen for every archer. You will lose more units if the city is built on a hill, or has other defensive factors. However plan for the worst, have extra units. luck is a big factor in civ games.
4. As the game progresses, city defenses become more powerful. Luckily, siege units become very powerful as well. First, bombard down the enemy defenses. Then suicide siege units until all the defenders in the city are badly damaged (artillery causes collateral damage which damages multiple units in a stack at a time. You then should have excellent odds with the rest of your offensive units. You will go through lots of artillery but your troops will become veterans and receive many promotions, if you keep upgrading them you will have a huge advantage.
5. Finally, only keep cities that will benefit you. Useless cities drag down your economy hugely. Keep either very strategic cities (such as one that blocks off a land bridge) or profitable ones. City placement is very confusing when you are a beginner, but generally try to keep cities that have lots of resources, and have lots of hammers, or lots of food/commerce.
6. When you consolidate you're conquests make sure to build courthouses to reduce maintenance costs in your new cities.