The most important thing a new Alistar should learn is the WQ combo. You should master it, as there is nothing more annoying than chasing an enemy and your Alistar knock him away. The trick is: Just as the Headbutt animation starts (once you start moving towards an enemy), press and holdPulverize. Don't let go of it, because once the headbutt ends, Pulverize will automatically trigger.
This tactic is risky, leaves you useless while both abilities are on cooldown, but is considered a staple move of Alistar and used very often in both competitive and casual play.
You can use this to your advantage by using Pulverize followed by Headbutt to push them into your turret's firing range. Be aware that skilled players will know this gimmick and will not fall for it unless they are caught unaware.
You can use Flash to catch your target off guard, followed by Pulverize/Headbutt to push them into your allies/turret.
Also you can use Flash after Q instantly and catch enemy faster with animation declining
In the event that a tank is running close to your target to protect them, a safer and often more useful strategy is to use Headbutt on the tank to get you into range to Pulverize your true target.
Be wary when using Headbutt because you might end up disrupting positions in a way that benefits the enemy team, particularly knocking them out of a teammate's area of effect.
Alistar is considered a great roamer due to the fact that all he needs is his Pulverize and Headbutt combo to gank a lane.
Hitting an enemy into a close-by thick wall with Headbutt will stun them for ~1s, because they will still try to travel the distance, though the wall is in the way. Thus they will be stuck against the wall for the duration of the flight.
However, if Headbutt is used near a wall, it sometimes sends the enemy onto the other side, no matter how thick, allowing them to escape.
This can also be used to your advantage. Sending the enemy tank or a high damage dealing carry (like Fiddlesticks after he uses Crowstorm) over a wall can allow your team to win a 5v4 match.
Most walls in the jungle and river are thin, but whether they will go over them or not depends on how close they were to the wall at impact. If they are a high priority target, it may be better to keep them in the fight. A high priority threat would be neutralized if knocked over.
Unbreakable Will should mostly be used when turret diving or in team fights when you are being focused.
Try to buy a Sheen to increase your damage and pushing capability.
Sheen can then be upgraded to Iceborn Gauntlet for extra disruption and survivability in teamfights (in addition to the armor, mana, cooldown reduction and collateral splash damage all help with letting him heal more often) or Trinity Force for damage and mobility.
When you are in your turret range, consider staying close to Alistar so that if he uses his Triumphant Roar he will draw turret aggro because of his passive damage from Trample. But be aware that would make you an easy target for a Pulverize->Headbutt combo.
Flash can be used on Alistar to quickly get into the middle of a team, hit much of them with Pulverize, and knock a squishy target toward his team with Headbutt. Always spread out when you see Alistar walk toward your team.
Beware the potential combo of cancelling the knockback of Headbutt into Pulverize. At that note, it is safest to poke Alistar with something that is farther than the 650 ability range of Headbutt.
Alistar is a pure tank that naturally has a high health, armor and magic resistance. Buying items such as Last Whisper or Void Staff can help with killing him faster.
It is best to wait until the duration of his Unbreakable Will wears off before continuing to attack him.
If Alistar has already cast Pulverize and Headbutt, consider ignoring him if possible for the cooldowns of those two abilities are very significant - especially since spending effort to attack him may just have him reduce it with Unbreakable Will.
Due to said significant cooldowns, crowd-controlling his own team as he uses Pulverize if possible can waste the ability's synergy with his allies and leave Alistar with little else to do for his team while it cools down - as long as you yourself have other things to use while you traded cooldown(s) of your own for his, of course.
Avoid fighting near minion waves to ensure that his Triumphant Roar has a long cooldown.
Position yourself in such a way that it is difficult for Alistar to Headbutt you into a bad position that may endanger yourself.
However, be wary not to get in range of Pulverize instead. Try to position yourself properly and keep distance at the same time.
Flash or other movement abilities (such as Arcane Shift) can help make it difficult for Alistar to land a combo on you, or make for an easier escape after he already has.