I recently started playing Rakan and I quickly fell in love with him. I looked up some guides and most of them told to build Ardent, Redemption and Solari for the first three tiems. But ive seen Rakans starting with and building other items. What is the best way to build Rakan? And what should I be building/using in certain situations?
nothing situational, build ardent and you win 
or
45% good tankyness, strong shields, strong heals. sometimes ill buy mejai's for scaling, since rakan has the best scalings of any champion and in really rare occasions ill ditch locket and knights vow for 
and get around 600-700 ap
Rakan is a pretty versatile support. Some play him like an enchanter others play him like a tank. He can make great use out of most if not all support items. It all depends on team comps and your playstyle. I have had more success with him recently with a tankier build. Locket into Knight's vow. As an enchanter Ardent into Redemption is great in most matchups. I prefer to get mobi or swiftness but the boots listed by Gaytrox can work.
TLDR: Rakan is an extremely versatile champion. Find what works best for you.
I preferred the enchanter route because thats who I usually play but you get bursted down really fast if yo do that so you need to choose your engage carefully (pretty much how you play mid lane). But when you arent going in just peel for your support and shield him for the ardent boost. For the tankier ones I see they just go in and become a distraction for the whole enemy team probably going to try it out sometime soon.
From my own personal experience, if you treat him as anything but a vanguard/enchanter then you're not using Rakan to his fullest potential. I've never lost against an opposing Rakan nor seen any of them as successful.
Rakan's strength is his mobility. He can cover a HUGE amount of distance in a short amount of time. So build around that. My build path goes as follows: 
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Start with coin, it gives you more gold than any other support item for your kit. I rush mobi boots because not only does that allow me to roam to mid lane & help them snowball, it gives you MUCH more of power and threat in-lane. The faster you can run, the better chances you'll have at: chasing enemies down and landing your W at the best moment, punishing your opponents for over-extending, zoning, landing your ult on an entire enemy team, reaching your teammates in time to shield them from danger, etc etc. After mobi boots, i finish talisman of ascension. The BEST benefit of this is the engage potential. If you coordinate a gank or engage, activating it gives your team a 40% speed boost to keep up with you and jump on your STRONG AF engage. It's absolutely game changing and anyone that builds does not have as as strong of an engage. Vision is obvious, then ardent censor. This item is op right now, it also pretty much offers everything Rakan wants: AP, CDR, + Heal & Shield power plus the buff to your teammates, and movement speed. Redemption is a great item to have since Rakans heal & shield is limited. Then the situational items: Knights Vow and Zeke's Convergence. Knights Vow is great at keeping you and your marked teammate together. If I have an ADC that lacks disengage or isn't doing too well, I will build this. You and your teammate gain +15% movement speed when moving toward one another which lets me catch up to them if they get dove by the enemy, and lets them catch up to me when i engage the enemy team. If my adc can kite well or is just fed af and doesn't need me as badly, I'll usually build zekes convergence and focus on doing more damage instead.
*Solari is a decent item but not necessary. Rakan already has great heal and shield power for an individual teammate so it's just overkill. If your adc gets dove by a zed or gets pulled by blitz/thresh, all you have to do is W the enemy to keep them from capitalizing on the damage, q them, shield your teammate & heal them in the process because your Q instantly reactivates, then proceed to punish them because you have more cc and peel potential than most.
Rakan is a slippery mofo. You can dive in the middle of 5 enemies and fly back to the safety of your teammates before you even get touched. There's NO REASON to build tanky items on him. The only reason anyone would need to is because they constantly fail to land their W and get punished for it.
Combos: W-AA-Q-AA-E-E. Basic combo before level 6, great for punishing enemies who overstep. Engage with your w, in the time the enemy falls back down from the knock up you can aa once. Q them to start the heal charge, aa them again, then E to shield your teammate from any damage which also procs the heal. E them again since you can reactivate it to create another shield. AAs maximize your early-game damage. R-W-Q-E-E. Combo for teamfighting. If you can fit in some aa's, then thats fine but not necessary. At the point of grouping/teamfighting you're really not going to be doing too much damage with them so focus on landing your engage & protecting your teammates. *WHEN YOU ACTIVATE YOUR ULT: Hit as many people as possible. The effect lasts long enough to get an entire team charmed, and once theyre charmed theyre going to be walking toward you which lets you land a HUGE knockup on a full team with your W.
If you're chasing the enemy team and you're behind your teammates, use your e to cover the distance and catch up to them. If you're just too far behind, pair your flash with your ult. R-flash-W. It will catch anyone off guard and it makes it 10x easier to charm/knock up an entire team. Be gutsy with your plays but don't forget about peeling & protecting your carry. Max your w first, then your e, then Q.
Sidenote: I have tried multiple builds and lanes with rakan. HE DOES NOT DO DAMAGE. Even if you build him full AP, you're not going to be that strong. I HAVE won mid-lane vs a yasuo, I've won lane against a nasus as well. I've lost against a Katarina. It obviously depends on the matchup but Rakan is going to have a tough time versus anyone. The only benefit toward playing mid lane was the stronger roaming potential. That being said, since he can't trade well and lacks strong waveclear, that means sacrificing midlane to help your teammates snowball instead.
Tailisman into ardent, locket, sightstone, righteous glory is how I play him.
