For solo queue, and assuming that bronze = beginner/amateur. I think what suits this level of play are champions with lower tech's, have tolerable counters AND mechanics that transition well into more "advanced" champions, but I'll wait for recommendations =X

Top: I'd learn yorick, garen olaf. Everybody and their mother's seem to be playing garen so learning him would be good just so the other team can't have him. Yorick is good against many top laners that are used frequently and olaf while not ad effective as he used to be is still extremely viable and useful for team fighting and killing the adc.

Mid: get ahri she's easy high mobility good damage good sustain. Veigar is a guaranteed nuke for any ap champion and is easy to pick up. Kassadin is a bit harder to play but if you get good with him he's useful in nearly any team comp.

Jungle: nocturne and hecarim are both super good and their ults create some of the best ganks in the game. Good for.counter.jungling as well.

Support: taric and sona are a must at the very minimum. A good taric is huge and can protect very well with his stun and tankyness, sona ult and poke and heal are great in the right comp.

Adc is up in the air: I'd say Caitlyn is good to start off with, and mf seems to be super good early game and doesn't drop off too hard late game

Hecarim is not that beginner friendly. This is why everyone thought he sucked on Release... "He's Skarner, except worse!"
There are tons of tricks and builds for him. Jarvan is with a tiny bit of Ult practice, Olaf too.

Ashe is a good ADC to start off with (an arrow in the general direction of the enemy team can win games), as well as Tristana (she farms without even trying) and Sivir (this will teach you mana management like you wouldn't believe)
Cait also has those little tricks that separate a good ADC from an absolute feeder

Supports are bang on.

Learning top definitely goes to Garen, he's very straightforward.

Annie mid is also another basic burst caster to learn.

Anivia might be a nice mid to try out. She's strong on her own, and learning her early makes the other APs a cakewalk to master.

For top, don't forget Riven. Her mobility can make recovering from being out of position easier and not lethal. I also suggest learning Garen (as the previous posters mentioned).

Bot ADC Fotm seems to be either Caitlyn or MF but don't forget about Ashe, Tristiana or Graves too. They aren't the strongest choices right now but in the right hands can still be effective.

Jungle could be wide open. Nocturne is the FotM because of the S3 reworks and the tournament picks lately. Hecarim is a strong tanky carry late game. Xin Zhao is strong on initiations (ganks), Sejuani is getting a bump in popularity due to the new League of Warmogs and her W scaling on max health. Those are a few examples.

Hope this helps.

Easiest junglers to learn and do well with; warwick, fiddlesticks. both have near infinite sustain in the jungle.

I'll also add AD yi to that list. with the buffs to base heal on meditate he can farm the jungle extremely well and is fast enough to counter jungle in his regular path. his highlander ganks are also hilarious.

for learning mid, its a tough call cause mid is in flux ATM, lots of AD mid cropping up to stomp low levels (think panth, talon, khaz, etc). so I don't have recommendations there.

for top lane, garen is easy as hell and you just have to avoid getting countered by a teemo.

don't very familiar with bot lane, but taric is an easy, versatile support to use while learning.

....I have Garen, but the things he talks about as he moves really annoys me. What about Darius?

Pick anyone and use them until you get to know them really well. No one mentioned Kog for ADC because he might not be the easiest ADC to learn, but he is my main because I've used him a lot.

You hear people call certain champs OP because maybe someone had a bad match up. I don't think anyone is truly OP but sometimes there are bad match ups. Learning counter-champs takes a long time and seems to change with every patch. So, while you are learning whatever champ you want to for each position, don't expect to perform the same way in each game. Learn which opponents your champ does well against and with in each game and vice versa. I love playing Heimerdinger. He does well against mids like Ahri, Ryze, and even AP bruisers like Katarina but doesn't do well against Syndra, LeBlanc, or AD casters like Talon and Zed. So, even though I like playing Heimerdinger, I don't recommend him to other players. It's up to you to figure out what is good for you.

Try all the free champs during every rotation. You may find a champ that feels perfect for you.

Darius is a beast top laner.
Vi for jungle.
Kassadin/Ahri/Swain for mid
Ashe, Sivir, Varus, Ezreal for ADC
Sona(will teach you how hard it truly is to support), Taric and Soraka are the current OPs.