If you're starting out, I suggest you buy a variety of (cheap) champs. There's really no point to making one champion play two roles.

Have a look into Lee Sin. I'm sure others can give you a plethora of information regarding other multi-purpose champs but in my experience the most versatile champion in the current meta is Lee Sin.

I have to warn you, he requires an incredible degree of skill and proper positioning in teamfights to utilize his effectiveness. He isn't great at solo mid because he has a tendency to get zoned by AP carries who usually pack a lot of hard CC and burst dmg. His kit is usually most effective against bruisers and AD carries because of his E. As a solo top champion he's excellent, with the proper ability to harass and retreat to safety, and his has excellent coordination with the junglers for great ganks. However, his best role is as a jungler because he has great sustain, jungle speed and ganking ability, and his mobility allows for strong counter-jungling.

He is built usually as a tank or tanky DPS, able to dish out tons of damage and take a few hits. However, he is a very combo-dependent champ. You need to learn the proper combos in order to maximize the usefulness of his passive but also at the same time maintain proper control of the battle. His ult is incredibly useful in teamfights, able to isolate certain champions for the team and provide lots of CC.

TL;DR: Basically take a try at Lee Sin. He's difficult to use but incredibly rewarding if you manage to understand the flow of his abilities.

Nautalis.
Can go top, jungle, mid, support bot.

Can build tank. Tanky ap. Or glass cannon ap and still have good damage.
Need to position yourself to so as to peel off and protect your carry from damage.
Fits in the Meta.

Malphite - Can play AP mid, jungle, top. Not sure about support. He's also really strong late game... and an unstoppable force if fed.

GP - Jack of all trades, master of none. Can play pretty much any role, but doesn't excel at any of them. His kit can do a bit of everything, but doesn't do anything better than another champion with a similar skill. However, he is a pirate with an awesome laugh.

Lulu - You can build pretty much anything on her and she can still function well. Her utility/kit makes her fall into a supportish role no matter how you build her, but she can function well in pretty much any lane with many build paths.

Nidalee - She's like a Swiss army knife. Her E serves as a mini-heal and an awesome AS steroid. Has traps that can serve as many wards and lower armor/mr of any enemy that steps on them. Can do massive damage with her spears if built AP, her Q becomes a nuke if built AD. Is incredibly fast, hard to kill, easy to escape with (she can hop walls), and can destroy towers incredibly fast. I don't like the way she looks though, her boots especially. If her appearance got an upgrade like Ashe, I would probably use her more.

Lee Sin - His kit let's him do anything on the map, when he pleases. Tons of chase, tons of escapes. Fair and balanced shields, lifesteal/spell vamp, and energy resource ensure he never has to leave the lane/jungle, unless he chooses to.

Jarvan - This dude right here. Can go in any lane and has incredibly versatile build paths. Has a gap closer that can go through walls, and can escape ganks and/or chase someone down efficiently. Becomes a raid boss if he gets fed. Can 1v3 while building pure tank np.
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In general, champions that can easily jungle as well as lane, and can build to counter the enemy without being set back damage wise, tend to be fairly versatile.

kennen

Let us look at Shen.

-Solo Mid - Nah...

-Solo Top - Very good - banworthy

-Jungling - Doable. There's better, but he's good at it.

-Support Bot - Very doable. Some might ask "BUT WERE IS TEH HEAL?" Q is the answer.

-AD Bot - Sadly, no.

-Tank - Yes!

-Fighter - He can output some damage, but isn't the strongest.

-Assassin - No, sadly.

-Requires proper positioning - Only for hitting Shadow Dash.

-Requires proper patterns (such as maxing out Ryze's cooldowns or taking advantage of Sona's Power Cord) - Ehh... Not really.

-Viable in the current meta - Hell yes.

-Viable outside the current meta - Hell yes.

-Does not cause people to queue dodge if seen - HELL yes. See the pattern?

-Does Phreak use this champion to get first blood, use it with a trinity force, and jungles with this champion? - No.

-Champion is good on Summoner's Rift, Twisted Treeline, Crystal Scar, and the revamped Proving Grounds? - Good on EVERYTHING but Proving.

-How likely is the champion going to get nerfed (I do not want to use a champion that is going to get nerfed every other patch)? - He just got his buffs/nerfs. He's not going to change for a long time.

-Is it commonly used (I do not want to use someone like Teemo, whom everyone uses...all the time) - Semi-common, but good Shen players are very rare.

Nidalee fosho

Oh, and Kayle.

I just hit level 30 right before the most recent patch. However, I recently felt like tacking a long term project that would require me take a lot of time and practice in order to become sufficient at everything the champion has to offer.

I have at least two champions for every role.

These are the type of champions I am looking for. A versatile champion that requires time and practice to get good with.

And of course, SaveTheMaids and Vriska Serket, your thoughts are also very appreciated. And Mortamor too.