At which levels does Yasuo get his power spikes?

It's somewhat subjective depending on the enemy team and your own team.

Yasuo's first powerspike comes from leveling up his E (which tends to happen at level 2 or 3 depending on who you're facing).

After that Yasuo has a powerspike when he reaches level 6 (due to being able to cast Last Breath).

Depending on his team, he also gets powerspikes when he is with them and they get reduced cooldowns on their own knockups (Malphite's R, Cho' Gath's Q for example)

Finally, Yasuo gets major power spikes at levels 11 and 16 (But at this point item powerspikes are more noticeable) when he can upgrade his ultimate:

COOLDOWN: 80 / 55 / 30

Physical Damage:200 / 300 / 400 (+ 150% bonus AD)

Meaning that each upgrade adds 100 flat damage on top of almost halving the cooldown (or doubling the ammount of times you can use your ultimate)

A more general answer:

Almost all champions experience similar powerspikes when it comes to levels. For items, it varies a bit.

I'll use Yasuo and Syndra as my main examples (Yasuo since you mention him and Syndra for the passive dynamic) with some other champions sprinkled in.

Level Power Spikes

Lvl 1 - 3: this is the most dependent since you usually unlock your entire skillset in these 3 levels. Characters that are casters (usually mages and a few ad casters) experience their first spikes at lvl 2 and/or 3, depending on what combos they unlock.

Example: Syndra at level 2 now has a combo as long as she has q. Yasuo also gains a huge boost in mobility/damage with his q/e interactions. At level 3, Syndra has another spike since she can get a larger combo off with more spheres, while Yasuo doesn't spike since he is either leveling up a skill for more damage, or grabbing windwall for defensive use. Another example is Leona and Alistar. At level two, they get their full CC combo and become quite intimidating. Then there's Riven at level 3, who can now fly around lane with either a damage, shield, or stun whenever she wants.

Level 6: Obvious powerspikes are obvious. You get your ult and that's a huge spike.

Level 9: This is the first level when you can max a skill. This is partially a spike for some characters, and for others, it signifies a slight decline in their level scaling. They won't experience as may raw spikes, but their damage in general will increase as the skills scale off each other.

Example: Syndra gets a bonus when a skill is maxed, so hitting lvl 9 can be a spike for her. For Yasuo, you have maxed a skill out as well, but now have to put points into a different skill. Vladimir has maxed out Q and likely has enough sustain to harass you in lane decently. You may not be stacking overpowered q's after a few levels, but now you have a faster CD on your e and can move more. So less relative damage, but it's more distributed with mobility, or vice versa.

Level 11: Another obvious spike. Leveling up your ult and getting the increased damage/cdr/utility whatever it is.

Level 13: This is the first level when you can max out two skills. For certain characters, this is where they really start to shine. Many champions have two offensive skills and a single defensive skill. Some champions have very good scaling, so when you can max out the two offensive skills you may find yourself to be very intimidating. This is getting towards late game where hypercarries start to shine, simply because their kit is strong.

Examples: Yasuo with Q and E maxed now has a lot of damage, and a lot of mobility. Syndra likely has strong spheres and either an extra slow on her w or extra disengage on her e. Someone like Vayne now has her low CD tumble and the giant true damage on her w.

Level 16: Last obvious spike with your final ult point. Syndra gets more damage, Yasuo has a low CD, everyone else has their thing. But it's a point in your ult. It's important.

Item Power Spikes

Items are a bit tricky since everyone can have different builds for different champions. But in general, here's what to look for.

High Efficiency Purchases: Particularly in early game, high efficiency purchases can get a strong edge.

Doran's items have great efficiency with the trade-off that they can't be built into anything else, thus requiring you to sell them for a loss later in the game. But if someone comes to lane early on with 3 Doran's items, they probably REALLY want the extra stats and intend to bully you enough where they make up the gold loss. (example: D Blade is 137% efficient)

BF Sword or NLR are both very expensive items, but they provide a better gold/stat ratio than their cheaper counterparts. If someone comes back with a more expensive item than you, they're also getting the benefit of that ratio. (example: BF is 107% efficient while pickaxe is only 100%)

Finished Items: When you finish an item, it generally comes with a similar gold efficiency as the components PLUS an effect on top of it.

First Items. When you finish that first item, if it synergies with your kit, then you tend to experience strong power spikes. For Yasuo, items with crit and attack speed synergies amazingly, so an early shiv will provide a huge spike especially since the item's passive works with crit and has a decent base damage. For someone like syndra, having extra orbs out provides her with more ult damage and options, so any CDR item will give her a decent spike. Building an early morello or athene's gives her CDR and enough mana to make use of it with her damaging increasing anyway due to her Q/R interaction.

Second Items. Finishing a second item is enough for a lot of hypercarries to being wrecking havor, with good reason. Hyper carries tend to scale extremely well off multiple stats, and few items provide all the stats needed. A second completed item benefits hypercarries because they now have all of the stats they need. Jax, for example, uses a lot of attack speed and all offensive stats. Combining these with his passive and ult, he also gets defensive stats and scaling damage for his skills. One item can't cover everything he needs (although triforce tries), but a second item makes him deadly.

For someone like Yasuo, getting a second crit item on him will lead to 100% crit chance. Completing a Shiv and IE will give him all the offensive stats he wants while generating two new passives THAT SCALE WITH EACH OTHER. This is why second full items are strong. It's when champions usually get their large damage bursts.

Other Items. After the first two items, it is even more dependent upon the champion and the matchups. Hypercarries will be able to get more stats and have more points in their skills and just scale like crazy. Squishier carries can start building defensive items so they can survive longer and kill more people. Depending on the build, you may or may not see additional powerspikes.

Most champions can snowball off the first two items, and the level powerspikes will get them through between the item spikes. Tanks are especially dangerous since defensive items are typically cheaper and most of them have strong base damages anyway. Please note that this is just a GENERAL guide and is highly subject to the individual champion and each game and matchup. Hope it helps.