Puck is a versatile ganker/initiator with a lot of options in any given situation. No guide can tell you when to use which skill, but the idea is to help you understand the potential that lies within the hero. Most importantly when paying Puck, don't forget that you are only 1/5th of your team and you aren't going to be dropping their carry like Lina or throwing down DPS like Naix. With your team, you can devastate team fights and play a key role in the game.
In this guide, I often refer to the skills as Orb, Rift, Shift, and Coil. It is written to be dense, concise and general, so some things may be left out, but this isn't a game where a ostentatious book about Puck will make you much better.
Strength: 15 + 1.7
Agility: 22 + 1.7
Intelligence: 25 + 2.4
Damage: 42 - 53
Attack Animation: 0.5 / 0.8
Casting Animation: 0.1 / 0.51
Missile Speed: 900
Attack Range: 550
Sight Range: 1800 / 1200
Horrible Strength, get Bracer over Null Talisman if you need the early HP and are okay with last hitting without the +6 Int. Also has low move speed, so boots and level 1 Orb are pretty important. You're very vulnerable to ganks early on, so avoid disables with Orb/Shift as well as you can.
If you don't know what the numbers on the animations mean, the first number is the startup time - if something is canceled in this time, nothing will happen. The second number is the time after the attack or spell that a hero animation is playing but it can be canceled by any command. You should cancel the secondary animation if you're experienced enough to not cancel the attack itself, but it's not as bad as some heroes (e.g., Crystal Maiden). This is most important when chasing someone down because it can help you get in a crucial extra hit by canceling attacks into movement.
There are two main builds, both starting with level 1 Orb and level 2 Shift. These two are picked first to keep you from dying, giving you both an escape and a nuke dodge. Mastering Orb + Phase Shift is the first thing you will need to do with Puck. The basic combination is Orb away from opponents, then Shift to avoid a stun / lethal nuke, then replace Orb before it expires and after it's far enough away.
The Waning Rift build in the first tab scales better and is more difficult to dodge, plus it provides a solid silence and protection against melee. You still have Orb to escape, but it isn't maxed first because its utility doesn't improve over levels. It's a better build when paired with a lane partner who can disable/nuke, but not as good for soloing middle as it requires you to get close to targets.
The Illusory Orb build can still be used in a side lane if you don't need the silence or aren't facing any melee. Additionally, Orb's damage doesn't scale as well as Rift but its mana cost doesn't increase, so if you're bad at managing mana it can be more useful. Orb can be useful to escape up hills or take shortcuts when chasing by going over impassable terrain.
Phase Shift is a unique ability that can prevent damage from any source. Manually cast, it lets you dodge stuns like Vengeful Spirit's, Sven's, Lina's, etc. On autocast, it can dodge instant damage like Bounty Hunter attacking you out of invisibility, but it can also dodge Zeus' Lightning Bolt if you're in Static Field range, Mirana's Arrow, and Spiritbreaker's ultimate. When you use it and whether or not you use autocast depends on your experience, the matchup, and what enemies are near.
Blink Dagger is vital because it combos perfectly with Phase Shift. Phase Shift gives you time to avoid being hit so Blink Dagger will not be on cooldown (it goes on a 3 second cooldown when you take damage from heroes).
Power Treads are usually unnecessary since you are going to be doing burst damage early on and you will want to get Boots of Travel later for the mobility and the global presence. You may want to get it if you have a strong pushing team with enough Arcane Boots and heals to keep you from needing to go back to base. Treads also can provide +8 Strength, but unless you have a very squishy team, you do not need them and should be avoiding damage as much as possible.
Ring of Basilius, Drum of Endurance, and Urn of Shadows are usually picked up on other heroes because you need to focus on getting your Blink, but they all work fine for Puck. If other heroes can carry them, don't get them. The mana regeneration and armor from Ring is useful but Bottle is better. Drum of Endurance helps you chase and run but you already have skills for that and are better off with Boots of Travel. Urn of Shadows is less important than Bottle + Null Talisman since you want to control runes and increase Intelligence over Strength for a better mana pool and a bit more damage.
Eul's Scepter of Divinity and Force Staff are usually picked up when you're getting silenced a lot, because they function as escapes when you can't use Phase Shift and your Blink Dagger is on cooldown from hero damage. They can function as luxury items as well if you don't have enough gold for the real luxury items.
The other luxury items are fairly self-explanatory. Ghost Scepter and Shiva's Guard provide late game defense against DPS heroes like Drow Ranger or Juggernaut. Scythe of Vyse is just an amazing item that adds an additional disable to your team, so if you don't need or already have defensive items, get it. The situational items in Build 2 are very rarely needed and are better on other heroes most of the time.
Try to get mid solo, focus on last hitting for a quick bottle and getting the runes. Use orb to beat opponents to runes if you have to, and look for ganks whenever you can. Since your abilities are AoE, you can often get double kills if your team pays attention. Watch your opponents' reactions when you go to gank, because sometimes they give away whether they have wards or not. Sometimes you'll be able to get a kill at middle, but unless they have a carry there, it's less worthwhile than helping your carry get kills and farm or ganking their carry to shut him down. Also, you have 550 range so it's difficult to win harass battles middle if they put a 600 range hero there and/or your Orbs get dodged. Don't be afraid to ask your team to buy Observer Wards if they don't do it themselves - your ganks from rune control will give more than enough return on investment for you team mates.
If you're bot/top 2v2, you'll want to go for kills around level 5/6 if you're laning with a decent early hero, so harass and courier yourself regen items if needed. Last hitting and denying is tougher with 3 other people in the lane, so if your lane mate has better base damage than you, try to pressure your opponents when they go for creeps. If you're with a stunner or even a good slow, you can get close and combo Orb/Rift to get at least 1 kill. Other important things to do include pulling jungle creeps if you're in the proper lane, ganking their jungler if they have one and you're in the nearby lane, and helping mid get runes if needed.
If you're bot/top 1v2/1v3, play it safe, stay in the fog as much as possible, and avoid getting disabled using Orb/Shift. You have low base hp and you'll probably want to go for Bracers first instead of Nulls. Hopefully you avoid death long enough to get a bottle/boots, but if not try to get a Magic Stick and Town Portal Scrolls. You won't get many last hits wth your low base damage and if they're denying the lane so it doesn't push, just use Orb to get last hits when it's safe and you won't immediately need it to escape. Try to get your team to buy you a ward so you can stop them from pulling creeps if they're doing so. Once you're level 6/7, you should either ask for ganks in your lane or go gank another lane. You will be able to get your early items if you can successfully gank.
This is the prime part of playing Puck properly. You should be at every team fight all game unless your team fights while you're dead (which you shouldn't be in the first place). Usually, you will be the one starting the fights so you must have good awareness and pay attention to your team's cooldowns and mana. If you have a Tidehunter with Blink, he can go in first, but sometimes it's better to use Ravage a bit later since it has more AoE than Coil.
Before Blink Dagger, you will want to initiate with either Orb or Coil, depending on how far away you are from opponents. Try to initiate from fog so they can't see the Orb or you until it's too late for them to avoid it. Once your Orb is at a desirable position, take its place and use Coil immediately. Follow with Rift if you're close enough or can get close enough without being stunned, otherwise your team should already be coming in to follow up and you need to focus on avoiding death by using Shift or attacking if they're fleeing. The disadvantage of entering with Orb is that you don't have it for an escape (solved with Blink).
After you buy Blink Dagger it's pretty much the same, but with one key difference - you should never die. The simple combo of Shift -> Blink which will save you from nearly anything. The exact moment you should use each ability and the optimal time to initiate is something you have to learn by experience. Fortunately, your abilities have a low cooldown so you have plenty of chances to test the water before committing to initiation.
Your best team mates are any heroes that can follow up when you initiate in team fights or gank multiple heroes. Earthshaker, Tiny, Sand King, Tidehunter, or Queen of Pain can wreak havoc after you've disrupted their flow and caught them in a Dream Coil. Later in the game you can initiate along with a farmed carry and leave enemies helpless while they get beaten to death.
Silencer is pretty much anti-you. If they pick him, don't pick Puck or you're in a pickle. You can't chain your abilities around him, so if you use the standard Orb -> Coil you won't be able to Rift or Shift so they'll be free to stun/nuke your tail off. If you're the only one who can initiate well on your team, get a Blink ASAP so you can Orb -> Coil -> Blink out. Other heroes with silence such as Bloodseeker, Death Prophet or Drow Ranger are similarly problematic since Puck is all about his 4 active spells and has garbage base stats. Doom Bringer is nearly as bad as Silencer because his ultimate disables items as well, so you want to get a Rift on him and get out before he can Doom you (or have your team follow up on him).
Heroes with high mobility are also tough for Puck to face, but they also make great team mates that help you chase down runners. Examples are Morphling, Storm Spirit, and Queen of Pain.
Once you have his combos down and become used to playing Puck, there are nearly infinite ways to juke with him. With quick regeneration items like Bottle and Magic Wand you can cancel your secondary attack animation while juking to slowly whittle down your enemy's health and get your spells and mana back up to turn the fight around. Mastering the ability to turn a close engagement to your favor is one of the most high level plays in DOTA, and it's something Puck excels at.
You can also watch replays of high level games and tournaments where Puck is picked. Other than that, once you've read this guide, there isn't much more you can learn without experience. GL HF!