Given that League of Legends is nominally a "free-to-play" game, everyone needs to make their own purchasing decisions. But for the newer players, there are a few champions that everyone should be comfortable with. These are champions that are cheap, come up frequently in the free-champion rotation, and are reasonably good enough in their role that they still see play at most levels of play. For the most part, these are the simpler champions to play and to understand, which makes them good investments for beginners as learning tools.
Amumu (585 RP/1350 IP): Currently he's being perma-banned in rank at everything below platinum (and frequently even there), but this will change eventually. He's a powerful jungler with a single skill shot that's good to practice with. He builds tanky, so he's somewhat forgiving. This champion will improve your skill shots, positioning and judgement on initiating.
Ashe (260/450): She's incredibly cheap and she's many player's first ADC champ. Her kit is relatively straightforward, and she combines a large AoE slow with a powerful initiate. She is probably the most "even" ADC that I've played with. She's decent early, mid- and late game (in contrast with some champs, like Caitlyn, who start out strong, are weak mid-game and transition back to strength late game). This champion will improve your positioning, target selection and last hitting skills.
Garen (260/450): He's a frequent top-lane champ who occasionally sees play mid-lane or as a "support" in a bottom kill lane.Most Garens build incredibly tanky, making it useful for learning to play as and against a hyper tank.
Jax (585/1350): Along with Master Yi and Darius, Jax is part of the "Triumvirate of Noob Crushing". He tends to build with a mix of damage types and tankiness that can't easily be countered. Throw in a strong gap closer, stun, and counterstrike, and he's very difficult to play against if you don't know what to watch for. He's a good champion to start with early and will be a frequently seen champion at all levels of play. Even high-level professionals still play him occasionally.
Malphite (585/1350): Tanky top laners are a pretty common sight right now and Malphite fits that bill well. He has a great initiate that will likely become your team's initiate and has the added bonus of being super-tanky while still getting enough damaging abilities that he's not easily ignored. Armor scaling on his E, in conjunction with an Abyssal Scepter, brings a lot to a team fight. He'll really improve your initiation judgement, melee last hitting and mana management (he's a mana hog early game). He's another frequent ban in mid-ELO play, but I expect that to taper off as time passes by.
Master Yi (260/450): Another "Noob Crusher", Master Yi is very powerful against weaker teams. He's very strong against players who don't realize they need to carefully use their CC and also against players lacking in familiarity with some of the basic mechanics of the game. He'll also teach you how to split push/backdoor the enemy base which is an important situational skill to learn.
Nunu (260/450): Nunu can play support, which is the typical position at higher levels of play. With consume, however, he can also jungle before you hit level 20 and start getting the good runes. This can allow lower-level summoners to begin learning the jungling basics while also having a very good support champion.
Ryze (260/450): He's a prototypical mid-laner who still gets frequently played at high levels. He builds differently than most other mid-laners, with a reduced focus on AP and an increased value of mana. As a result, he's somewhat forgiving for newer players who tend to spam abilities a bit more aggressively. He's also tankier than a lot of other mids, so your mistakes are less punished there as well.
Sivir (260/450): She's one of the two best pushing ADCs in the game. She's relatively easy to last hit with and has the ability to wipe whole waves clean as well as zoning out opponents. Her spell shield is the big skill test here and you'll learn how to watch your opponent and time it right. Against champs with reasonably predictable abilities, (think Ashe, Urgot, Miss Fortune, or Caitlyn) her spell shield can provide her with significant amounts of sustain and allow her to push even harder.
Teemo (585/1350): Ahhh… Teemo. He's quick. He's annoying. He's fun. He builds a completely different way from other champions, so you get to learn a little bit about that type of hybrid build (Nashor's Tooth = good). I like Teemo as a beginner-friendly champ, because his mushrooms also teach you the power of wards. Play two games as Teemo. Then play a game without wards, and you realize why people buy wards.
Veigar (585/1350): He's a fun mid-lane, AP-heavy champ. He's really powerful against slower, shorter-range AP mid-laners (think Cassiopeia). His ult scales off your opponent's AP, so it can do massive amounts of damage. This is another champion who will teach you the power of last hitting and train you to be good at it. His skill shots require some tactical understanding, but they're not particularly "hard" to position correctly.
Warwick (585/1350): People rarely play him in the jungle anymore (which is funny because, along with Fiddlesticks, he was one of the champions specifically called out for a return to the jungle). He has good single-target damage, sustain and a powerful initiate. I put him on this list because he can teach you a lot about transitioning from a powerful ganking jungler to a split pushing machine. Since several junglers fall into this camp late game, this is a very important skill to learn.
Combined, these champions cost a grand total of 10,800 LPs. You should be able to accumulate these reasonably quickly if you simply play a couple games each night. As you find champions/roles that you like, you can always buy more champions to specialize in those roles. But this base (or something similar) will give you backup options for when and if you have to play a position that's unfamiliar.