Open your client and find the Champions tab under your profile. Make sure it's showing all champions. Now click the Recommended checkbox on the left. These champions are some of the easier ones to reach a competent level with. The carry champions offered focus on ranged carries, try Master Yi if you want a melee one.

Mastering any champion will take time because you need to learn the match-ups versus other champions. Once you find the play style that appeals to you, perhaps by observing your opponents play, stick to learning that champ. Forum guides can give you a good start on decent builds but you'll probably develop it your own way over time. When you're learning, keep in mind that switching to new champions won't magically make you better - unless it turns out that the champion just can't effectively do what you're trying to do with them, but that's the play style thing.

Oh, and if you try Annie, the white smoke thing around you means your next ability has the stun from your passive loaded. Other players can see it too. (The innate passive that every champion has sadly isn't on the ability list so you'll have to read it ingame or at leaguecraft.com or something. Don't rely on champion, item or rune information on official site because it's outdated.)

blitzcrank or twisted fate. the age of man has come to an end.

nunu is good and cheap. I say he is probably the most top tier champ of all the 450 ip champs if not ash

The 'difficulty' rating is something of a variable guide. One of the first champions that I started Not Sucking with was Cho'gath, who has a pretty high difficulty rating. There is a lot of personal taste and mindset involved.

That said, some champs that I wouldn't recommend playing if you're still starting out.

1. Evelynn. While Evelynn is a major threat, sure to make most players feel a spark of fear, whether rational or irrational on seeing her on the loading screen, she's a ***** to play well and very unforgiving of mistakes.
2. Kassadin. Some may take issue with this one, but seeing as Kassadin is the single-worst champion I've played (2/11/2 in my last game, whereas I can usually get at least equal KDR even with a champ I'm not familiar with), there is definitely something that takes getting used to with him. If you can get him to work, more power to you, but I would by no standards consider him noob-friendly.
3. Katarina. Kat is in much the same boat as Eve, except with a bit more flexilibity, more than one attack, and no stealth. A squishy melee-range assassin will never be easy to play, though I'd say she's one of the easiest of that category.
4. Singed. If you're still learning all the tricks of the game, the last thing you need is someone with a completely unique play style and slightly jarring typical build giving you a headache.
5. Twitch. Twitch is, I believe, the single squishiest champ in the game. His damage output can be staggering, and his abilities make a really horrifying little packet, but if you don't want to die a lot, you need a solid grasp of positioning, good knowledge of the opposition, and a decent gut instinct for when to engage and when to slink off. If you can do those, mind, you can get ridiculous KDRs unless your opposition knows what they're doing - which they won't at lower levels. Of course, he's also ****ed expensive.

The characters I'm currently cycling through, if it helps...

1. Cho'gath. First character I played, and the first I unlocked. I find that games with Cho can either be mediocre or awesome, depending on how intense and regular the fighting is and how big I'm allowed to get. He takes some getting used to, particularly the delay for Rupture, but he's a scary force if played well.
2. Tristana. This adorable midget is the second champ I unlocked, and she's possibly my favourite. She's a really nice character to build up with, as her range and relatively simple skillset make her easy to start out with and forgiving of mistakes, but the more you get a feel for her, the more deadly she becomes. It's come to the point where I can almost shut down my opponent in the middle lane unless they really know what they're doing. Of course, the rest of the team makeup doesn't always allow for a ranged carry...
3. Nunu. Not unlocked, but considering it. This guy is just ridiculous - at low levels, he's a huge pain in the neck to lane against, and at high levels, his damage output is staggering. A mid-level tank that can do 1500 damage to everyone around him, and hurl 500 damage slowing nukes every 6 seconds. Need I say more? However, he does benefit from an aggressive mindset.
4. Veigar. A recent flirtation that probably won't continue beyond his being free, but he's a nice character. With a little skill, you can quickly build up enormous amounts of damage, and Event Horizon has so many uses. He does suffer from being average except in duels and ganks, in my opinion, but definitely fun.
5. Nasus. Only just unlocked, still experimenting with, but he's underrated. Siphoning Strike allows him to get a pretty decent amount of damage if he goes tank, and a huge amount otherwise, while his ult can allow for some scary feats - I've nearly killed an enemy Sion before going down 4v1. Does require some skill with last hits to really meet his full potential, though.