Master Yi strongly prefers to Full Clear with the goal of maximizing his farm over time. Because Yi doesn't particularly care about early ganks, you will often find him starting on Bot Side to get the two man leash. More experienced Yi players however do have a preference towards Red Buff as Blue Buff does virtually nothing for his first clear. Others will employ early vision or Reverse Clears in order to try to avoid the enemy Jungler.
For most Junglers, it's often a good idea to try to follow Yi. If he paths from bot to topside, you yourself should clear topwards to meet him on the topside scuttle crab. Most meta Junglers can fight him right off the crab, limiting his farm and opportunistic ganks. More aggressive Junglers can also 3 camp clear and invade Yi at his second buff, often scoring an easy kill. In this, early vision is key. Although leashes are enough to know where he started, it's too late by then. You would want to already be starting on the same side buff as him to match him. If your team is particularly cheeky, a ward on the opposite buff can spot Yi doing a 3 camp/Reverse Clear. For the sake of brevity and relevance, I will write a separate article on how to counter Reverse Clears. Passive Junglers could path opposite of Yi, but the problem is that Yi outscales virtually all Junglers in a one to one comparison. It won't exactly matter to match Yi's farm if he makes better use of it. Certain powerspike Junglers like Tri-Force Junglers may be able to take advantage of such an opportunity, however.
Of course, Yi is one of the best scaling Champions in the game. Seeing Yi on the enemy team is like looking at a time bomb waiting to go off, and you need to either finish the game before he goes off or set him or his teammates far back enough to push the timer back.
Luckily, it is quite easy to stomp on Yi's toes in the early game. It should be noted that Yi actually statchecks a lot of champions in the early game thanks to his Passive, especially if he Alpha Strikes an ability, so unless you're playing a specific counter to Yi like Udyr, it's not always smart to initiate on him Level 3. He is however particularly vulnerable while he is farming for his first few clears, leaving easy counterjungling opportunities on the table. He also has difficulty initiating a lot of fights; can't Alpha Strike abilities if he used it to initiate. Often if you just start hitting the crab while saving an important cooldown for Yi, he can't do anything about it. In fact, you can take this concept even further; you can scare Yi away just by walking at him menacingly. Of course, it helps to play a champion that can duel Yi early, but even if he bites, that means you can unload your abilities on him without having to worry about Alpha Strike.
Yi's lategame win conditions are Hypercarry (Protect) and Splitpush. He can achieve both jobs reasonably well, switching to Splitpush against peel heavy teams and Hypercarry against more aggressive teams. Although both variations of Yi use Conqueror, you can identify a dedicated Splitpush Yi with Lethal Tempo. This is because Splitpush Yi loves stacking Attack Speed items, often overcapping when he Ults. Yi does happen to have many specific counters to keep him out of the side lanes, for example Tryndamere, Jax, and Kha'Zix. Otherwise he's pretty vulnerable to two mans as he's quite squishy for a splitpusher. Hypercarry Yi on the other hand likes to build a little Tankier. He'll usually start with Kraken Slayer into Guinsoo's, leaving him vulnerable to focus fire earlier in the game, but he'll often complete his build with items like Sterak's, GA, Death's Dance or a Lifesteal item. It leads to this dangerous scenario where he'll sweep teamfights if he gets his first reset, but it's also hard to actually kill him before he gets his first reset. In either scenario, the best way to deal with Yi is to force objectives early before Yi has time to scale, and have either a specific counter or a 2+ man team focus fire Yi to keep him down.
The single best item you can build against Yi is your Stopwatch. Both Guardian Angel and Zhonya's Hourglass act as fantastic stalls against Yi, giving opportunities to waste his Highlander and catch him out if he Alpha Strikes on somebody with the item. Unfortunately, that's as far as it gets as far as general build advice goes. The impetus is largely on Yi to correctly itemize for the enemy team, rather than the other way around. In this way, it's better to memorize your early powerspike items and start forcing objectives while on those spikes before Yi has a chance to catch up to you.