Pros:
Takes towers really fast
Fairly decent scaling
Destroys Tanks
Unstoppable when ahead
Unfamiliar match-up for many low elo players
Easy to pick up

Cons:
Really match-up dependent
Vulnerable to poke and CC
Very predictable and linear playstyle
Weak waveclear without Tiamat
Not the best at teamfights
Off-Meta (around 46% wr and 0.3% pr Plat+)

King's Tribute (Passive) is pretty self-explanatory

Chomp (Q) is your main damage source. You can also use it as an auto attack reset. Do not waste your Q in lane, since you can't really win trades when it's down. Q can be used on structures.

Frozen Domain (W) basically just gives you free stats. Always use it when taking towers, the attack speed from it makes Trundle one of the best split pushers.

Pillar of Ice (E) You can slow enemy champions and block their paths with this ability. It's really good for catching champions that don't have mobility spells.

Subjugate (R) Always use this ability on the most tanky enemy champion, since you get more value from it that way.

Precision

Press the Attack boosts your early game trading potential quite a bit. You can easily proc it with 2 autos and Q.
Conqueror As of 9.23 Conqueror no longer deals true damage so it isn't a no brainer against tanks anymore. However, I still consider it a viable option if you want some extra power for longer fights.
Triumph is always the best option from its tree.

Legend: Alacrity works really well on Trundle, since his damage comes from auto attacks. If you're not sure, take alacrity.
Legend: Tenacity is a must-have against heavy CC. Trundle does not have a reliable gap closer, so he is really vulnerable to CC.

Coup de Grace and Last Stand do pretty much the same thing. I prefer Coup de Grace.

Resolve

Grasp of the Undying -

Demolish helps Trundle get turret plates and split push later. If you kill your laner, you can easily get 2 or even 3 plates if the wave is in a good postion.

Conditioning is a really good synergy with Trundle's ultimate Subjugate, because the 5% bonus also applies to the stats gained from from Subjugate. The only downside is that the rune comes online after 10 min.
Second Wind helps you sustain through ranged **** & ball torture.
Bone Plating If you're not sure, you can never go wrong with bone plating.

Revitalize works well with King's Tribute and Subjugate.

Start Doran's Shield against heavy poke, otherwise go Doran's Blade.

I usually get Tiamat ASAP, but you can also skip it if you dare.

Grab Boots early, since you have no reliable gap closer.

Trinity Force is a huge powerspike for Trundle. Rush this item.

After Trinity I buy usually buy Titanic Hydra. It really helps with split pushing, and Trundle is great at that.

I believe that Sanguine Blade will be a really strong item for Trundle. It's exclusively an offensive option, but the passive has huge synergy with Trundle since his damage is auto-attack reliant.

After that it's all about situational items. Remember to buy Control Wards and Elixir of Iron if the situation calls for it.

Soon

Thank you for reading my guide!
Changelog:

20/11/2019: Updated for Season 10 and added changelog
23/11/2019: Added a brief matchup section