Firstly, if you're going to be fighting a lot of elephants or chariots, I recommend getting some phalanxes. Hire mercenary hoplites and stick them in your line to fend off such enemies. Second, make sure you have archers, and use them against the most worrying enemy units; chariots die rather easily against archers, and fire arrows can panic elephants. Furthermore, most shock units tend to have poor armor.
Otherwise, simply find the best unit you can get. If you're playing as Romans, triarii and principes pre-Marius; urbans post-Marius. Also, make sure to keep your general behind your line, and have him charge into the worst spots. Furthermore, make sure you have a good number of cavalry and use them to counter-charge enemy shock troops.

I believe Velites have a bonus against elephants.

I believe they do- and when fighting elephants, make sure your velites are in open formation.

Personally I prefer to have four units of foot archers per stack. I volley-fire them with fire arrows at elephants as soon as the beasts are in range. Nothing is quite so satisfying as watching pachyderms go apeshit and trample all over the nice, pretty lines of their own troops.

I suppose I could fill my armies with more skirmishers and archers. Phalanxes are a no go for me. I just don't like em. too cumbersome to command and besides, as Rome, recruiting and re-recruiting mercenary hopolites would be too tedious on account of I'm too far from greek and Ionic regions for it to be practical.
Till this point I had a standard army set up (too make recruiting armies quick and efficient) which relied heavily on not so heavy infantry, lots of cav, some dogs (love dogs), and only a little bit of archers. Worked fabulously against the likes of Macedon but now I'm too scared to go any deeper into the middle east.

Also I don't have a single city that can make urban cohorts.

My army as a roman general would at the beginning of the campaign be mostly hastati, equites and generals plus a few velites. Those are good enough to take Gaul, Carthage, Greece and Macedon. By that time you should have some principes, archers and some more cavalry, plus mercenaries.

No matter which family you play, from there on I use more cavalry and archers because:

Scipii face Numidians, who have mostly missile inf and missile cav, so I take lots of archers and cavalry to deal with them.

Julii face spain and britons, so take some archers to deal with chariots, you dont require anything special to beat the spanish. I find that barbarian mercenaries are good for absorbing chariot attacks, just make sure they dont break.

And in the east you'll be facing dacia, trace and pontus, and in my case usually egypt too. Dacia is best taken out with a balanced army with as core heavy infantry. Against Trace it would be wise to have some archers to deal with those annoying milia cavalry.

Pontus only good units, except for phalanx are all cavalry and mostly throw stuff at you, so keep some archers ready to hold them off and some cavalry of course.

Egypt has too much chariots to simply eliminate them with archers, so be sure to have enough infantry with good morale. And they have a lot of archers, including the excellent pharaoh bowmen. To deal with these and to flank the spear units have a large number of cavalry ready, but hold them behind your line untill the chariots are locked in combat, cause you dont want your cavalry running into their chariots.

Against Parthia you'll barely need infantry at all. Just some infantry to protect your archers and lock the eastern spearman in combat.

You really need combined arms in order to be truly successful in RTW (I'm working on an article about this). Going with an army heavily oriented towards just one type of unit is a recipe for disaster, as the Romans found out historically when fighting Parthia. Use more than a few archers and cavalry to compliment your heavy infantry as a Roman; I recommend hiring some Cretans if you can and recruiting Praetorian cavalry. Furthermore, for your primary army, make sure it has the best possible troops your faction can recruit.
Also, SOB, mercenary hoplites are available all the way from Scythia to Africa, and Greece to the Levant.

I have found that most of the shock troops (ie elephants and chariots.) need to move to be dangerous, so i muck them down with cannon fodder and then send in the good troops to finish them off. or, if I dont have any cannon fodder. hit them before they hit you. calvary is best for this

Welcome to the forums, Korbin.

I've found that those tactics definitely work (although I don't much like the casualties of the first once I'm later into the game). But I've definitely charged cataphracts into elephants before and won. Although one should watch for falling elephants, if your general is involved .

Also, SOB, mercenary hoplites are available all the way from Scythia to Africa, and Greece to the Levant.

Oh... I like to believe I know what I'm talking about most of the time. Thanks for the advice General.

hit them before they hit you. cavalry is best for this

I've tried charging cavalry into chariots before but they just seem to fall to pieces against them.
However, I will definitely try to incorporate more cannon fodder in my armies.

Oh... I like to believe I know what I'm talking about most of the time. Thanks for the advice General.

You're welcome . And I've used them extensively, with all different factions (from Scythia to the Julii), so I have a good idea where to get them.

I've tried charging cavalry into chariots before but they just seem to fall to pieces against them.

The lighter the cavalry and the harder the difficulty, the more likely this is; therefore, use only the heavier cavalries (legion and praetorian cavalry for Romans) against chariots, unless you absolutely have to attack. During my current Macedonian campaign, I used 5 units of light lancers and a general to kill three or four Egyptian chariots, and ended up with 3 men from the lancers and 13 from the general (out of 270 from the lancers and 55 from the general to start with). I did rout all of the chariots though.

It's easier to be the attacker on the battlefield rather than the attacked. If the latter happens they'll advance on you and you only have a limited time to kill the elephants and chariots before they reach you, but if you're the attacker then they'll more or less sit still and give your archers and skirmishers all the time they need to remove them from the equation.

For chariots I'd recommend getting all of your numerous archers to shoot at them (without flames as rapid shooting is needed) as soon as they're in range. If this is not enough and they slam into your line, try putting the attacked unit into guard mode so that they'll sit still and defend their ground - if you let them melee then their formation will break up, the chariots will wander all around them and causes losses, also they'll be vulnerable to further enemy units following them up. So with them in guard mode, acting as your anvil, you should have a few spearmen behind the line to charge in and swamp them. If you haven't got spearmen, more sword infantry will do, even your archers who afterall have nothing to shoot at anymore.

The same doesn't work for elephants unfortunately, they will just trample through whatever non-phalanx infantry you put in front of them. Your only real chance is to rout them before they get close enough to do damage. Put all your archers on fire arrows and try to get them to go nuts. Have your skirmishers forward too to throw spears at them at the earliest possible moment.

If you're fighting in a town it's very simple. Put your archers and skirmishers on the walls and send them hunting.

Two units with pila and velites to cover them should be able to rout elephants.

I prefer to be on defence, because it allows you to chooce the terrain. You can try to lure chariots and elephants into forest, or force the enemy to attack you on a hill, giving your archers an advantage, and tiring your enemy.

As Romans I dont really mind attacking either, but with a phalanx army, I hardly attack at all.

I prefer defending for the exact same reasons. I get to choose my terrain and can react better to the enemies movements.

Also, has anyone in RTW ever tried employing the same tactics to destroy elephants that Scipio Africanus used at Zama?

It doesn't work as it had with Scipio.

Opening the lines in quincux form only allows the elephants to get in then attack a flank, or to charge one unit and have its accompanying infantry (don't leave home without it) drive into the flanks.

Against elephants, you really need one of three things: 1) pikemen, 2) heavy cavalry, preferably cataphracts, or 3) incendiary pigs. The last are available to Romans, but do require a bit of skill to use, as too often the burning pigs run in the wrong direction.

Ballistae using flaming ammo work excellently against elephants. Found this out by accident the other day. One-hit kills... against elephants! And they went mad almost instantly!

I once managed to rout an elephant unit with a single shot of fire ammo by an onager unit. It was one lucky shot. However, I strongly recommend using phalanxes over missile infantry against elephants and chariots. Less mobile, but also less likely to be run over... As for incediary pigs, their use requires much practice and a bit of luck...

I've never used incendiary pigs before.
Also I don't like carrying siege equipment with me unless I'm planning to attack a city because they slow down the army.

I dont mind siege units in my army. Onagers can be useful to bombard enemies form a distance and help take out some elite units of their army.