I've heard the best way out of Bronze is to take advantage of unhealthy champs and carry your way out of it. I've chosen 2 champs that I like for each role and I will learn each champ properly before going into Ranked, let me know if they're good:
Top: Malphite/Darius Jungle: Xin Zhao/Rek'Sai (my main <3) Mid: Lux/Anivia Support: Nautilus/Blitzcrank ADC: Jhin/Sivir (already level 5 with her)
I wouldn't play Janna or Jhin in bronze. Janna because she needs a good ADC and Jhin because you can't rely on the enemy assassin(s) not being fed.
Well, you could try adding Nautilus top and jungle too... I'm having way too much fun (88 % winrate...) throwing anchors at people this season. ;)
Rek'sai is OP but half of her worth comes from the information and coordination you bring to the team. Unless you play with a focused premade, that kinda goes out the window in Bronze...
Janna has zero kill pressure. A great ADC will flourish with Janna in lane. Your average Bronze ADC will probably still play like they have Braum with them and go way too aggro.
Disclaimer: S5 S2-scrub, currently B1.
The best supports for climbing out of bronze are Blitzcrank, Morgana, and Leona. With any of these supports, you should play aggressively but carefully, focusing on trying to net your adc some kills. Janna is terrible in bronze. I have only once seen a good Janna player in bronze.
Top. Nunu.
Most top lanes in bronze are melee fighters who don't itemize or build resistances because "my build doesn't use it". Nunu is especially punishing because his snowballs are on a 5 second cooldown and as long as you're auto attacking, they're free.
His only problem is his late game presence (he has none) and his lack of escapes. Other than that he becomes an offtank later and his attack/move speed buff is great late game to throw on your carry.
Nasus is a Bronze DESTROYER. No one in Bronze knows how to play against him (takes his turret early then leaves lane, allowing him to free farm). If you want to consistently climb out of Bronze, look no further than Nasus.
Getting good with 
can win you just about every single game once you understand how to play him. He excels in roaming the map taking turrets which is exactly how you carry low elo games. You put so much pressure on the map that your team is always fighting 4v3. He has a different build path and skill order for every matchup. He's a good solo carry champion on low levels when people don't know how to handle a split push.
Try out playing Darius/garen top Yi and xin jungle(but use your main) Xerath and ziggs mid Sona and karma support Sivir and jinx ADC
Sona and karma do enough damage to replace a single laner Xerath pokes like hell and doesn't need much mana after a chalice, and ziggs has heavy waveclear and is pretty safe Sivir and jinx create tons of pressure in Lane, and have high support synergy(sivir in team fights is dumb) Yi and xin are just good at stomping And juggernauts op
I don't see how you can out of Bronze when your teams are so random and your duos are never online.
I disagree with some of these, but all of them can work really well.
Blitzcrank and Nautilus both rely on skillshot hooks, something not too good in bronze for the player, but great if you can actually land them since people are bad at dodging. Rek'Sai is great, but he is very team reliant. Jhin has a terrible winrate right now, and is pretty bad...
Oh yea btw. Super happy you play Rek'Sai. My own personal mains are Elise, Nidalee, and Rek'Sai. I would recommend watching Dekar173 play Rek'Sai. Look him up on YouTube. My personal Rek'Sai playstyle is a version of his.
Yi, udyr. Devourer champs are really good against bronze players
