Best builds for...
Top Lane
Top Lane
Top Lane
Top Lane 
What are the best builds for each one? Also, what's the best first items for Riven and Darius?
Top Lane
Top Lane
Top Lane
Top Lane 
What are the best builds for each one? Also, what's the best first items for Riven and Darius?
i think for riven you want to go either
or
+
x 3
For Darius, my first item depends heavily on the matchup. If the matchup is pretty even, i go
; if i'm facing a ton of harass like
, i go
+
x4 + 
Garen: Start with
and this will make it so you can stay in lane for an extended amount of time until you have enough gold to get either.....
First back:
and
OR
and
depending on the matchup
Necessary items:
(if the enemy team has no magic don't buy s.v.)
From then you can do whatever you think you would need from this list:

If you want any damage for some reason:

Example Build(in order):

For Garen I would say rush a sunfire cape first. It gives you some nice defensive stats, as well as some bonus damage for waveclear that also synergizes with your spin. After that, finish up your boots and build reactively to what your enemy has built, what their teamcomp looks like (do they have 5 AD champions?) and who is the most fed/who is dealing the most damage on their team. If they do a lot of magic damage, Spirit Visage, Warmogs and Locket are viable pick ups on Garen. If they do a lot of physical damage, randuins, thornmail, and Frozen heart are good items to counter them. Frozen heart, while you do not benefit from the mana boost, Is still cost efficient by 33 gold if we ignore the mana stat (and it makes you very tanky and gives a lot of cdr).
For Shyvana, Ravenous Hydra and the remaining items being tanky items like sunfire and randuins and warmogs work very well.
Quinn I don't play, but I'd assume you build like a standard ADC.
Teemo, rush nashors tooth, then you have a few choices. If your team needs ranged waveclear because the enemy is pushing into your base, Runaan's Hurricane actually works with his toxic shot passive and gives amazing waveclear for teemo. Liandries Torment and a Rylai's combo will make your shrooms hurt, Frozen mallet + Liandries will make your toxic shot hurt as well as Rylai's doesn't work on your toxic shot. Rabaddons/Voidstaff/Lichbane is nice if you are very ahead.
Both Riven and Darius do well rushing a Ravenous Hydra, but if you are behind as Darius, and are getting pushed into your turret, going tanky and farming under turret safely is the better option. If you are behind as Riven, finish your hydra, and keep building AD until you get back into the game.
You're better off going tank. If you want a damage item,
is the best way to go
Armor:
MR:
Boots:
or
Other: 
For
Damage:
and/or
. Triforce is for more burst with the sheen proc, and BortK is for more sustain.
Armor:
MR:
Boots:
or
Other: 
Build her sort of like a bruiser, don't go full damage unless you're extremely ahead, you should get defensive items so you won't get insta-killed when going into valor form.
Damage:
Armor:
MR:
Boots:
or
Other: 
You should go AP so you're shrooms will hurt alot.
Damage:
Defensive:
Waveclear:
Boots:
or 
You're builds should always be situational based on the enemy comp. These are just items that are good for these champs
There are three main options.
Full Tank - self explanatory. Sunfire Cape is probably the best tank item to rush, but build situationally after that. This is the best build if you find yourself falling behind or your team doesn't benefit from splitpushing.
Bilgewater->Full Tank - for more aggressive laning, at the expense of slightly lower impact in early teamfights. This is probably the most standard build for Shyvana, since it gives you some offensive options early on but doesn't compromise your tanking abilities by too much.
BotRK->Full Tank - this build is good in two situations. One, if you're against a Dr. Mundo (or any other passive health stacker). Two, if you get super far ahead in lane and want to abuse your lane opponent in a splitpush situation. Risky, but potentially very rewarding build.
I'm personally a fan of Tiamat, since it offsets her awful base HP/5 and makes her a lot more resilient to poke. If you're not having any issues with being poked down, triple Longsword into Brutalizer+Vamp Scepter can be pretty good, too.
Obligatory disclaimer about there being no best build for every situation, I would recommend in no particular order...
Garen: Sunfire Cape, Black Cleaver, Randuin's Omen, Maw of Malmortius, Locket of the Iron Solari
Shyvana: Sunfire Cape, Blade of the Ruined King, Locket of the Iron Solari, Randuin's Omen, Thornmail
Riven (bit more detailed): Brutalizer, Tiamat, Last Whisper, Youmuu's Ghostblade, Ravenous Hydra, [Defensive Item], Ionian Boots, Black Cleaver
Darius: Trinity Force/Ravenous Hydra/Sunfire Cape, Randuin's Omen, Banshee's Veil, Frozen Heart, Locket of the Solari