I usually play with friends so it's not really an issue. That said, you can get several champions dirt cheap.
First up, there's alistar and tristana, both of which you can get for free along with a skin through riots facebook promotions. That's 0 IP spent right there.
Second, the following champions are 1) Available for 450 IP, which you shoulf get in 3-4 games and 2) Considered decent, competitive picks:
Annie (the baseline against which bursty mages are usually compared)
Ashe (probably the best all-round AD Carry in the game)
Soraka (pretty decent support)
Poppy (high skill cap tanky dps)
Ryze (pretty popular tanky mage)
You can also get:
Kayle (support/carry. Kinda meh atm, but due to get a buff next patch)
Master Yi (pubstomper. Sort-of lulzy.)
Nunu (babys first jungler. Can also be played as support or solotop)
Sivir (carry situationally really good in certain team compositions)'
For both these champs, the free week champs and pretty much everythign else, always consider looking up a guide before playing them to get an idea for which skills are critical and what items work well with them. there are usually 3 go-to places for guides.
-- solomid.net likes masturbating to authors ELO ratings. The guides don't always bother explaining the reasoning behind their reccomendation, making them more of a quick reference than an actual way for new players to learn. Stick to approved/featured guides and you can usually find a cookie-cutter levelling/item build and a few pointers while waiting for the loading screen.
-- Leaguecraft.com lets pretty much everyone write a guide and relies on viewer ratings to sort the good from the bad. Always check when the guide was last updated since several of the highest-rated guides are many, many months out of date. They usually do a better job explaining champions for new players even if their builds are often suboptimal.
-- mobafire.com is an alternative to solomid.net. They have a few highly unorthodox yet surprisingly effective guides over there.
Again, I hasten to add you should probably play to level 30 without spending ANY IP/RP at all. Just play the free champs and learn which roles you like and if there are any champions you paticularly enjoy. Then, once you hit 30, invest your hard-earned IP in runes that match them and unlock them. If you decide you might want to stick with the game for a while after hitting lv30, consider grabbing a champion bundle. Yes, you have to pay real money for them. But it frees up your IP to get runes and the champions in the bundle are *heavily* discounted.
If those aren't enough for you, you can also get the following champs for 1350 IP (21% of what newly released champs cost):
Alistar (You can get him for free through facebook promotion. Support, jungler. Can be played AP too).
Amumu (Very good tanky jungler. Requires runes/masteries before his jungling becomes reliable. don't get until lv 30ish)
Cho'Gath (Mediocre jungler, situationally very good top lane. Can be played tanky/AP)
Dr. Mundo (Goes where he pleases. FotM jungle/solotop)
Evelynn (Don't get her)
Fiddlesticks (AP Jungle. Requires level 30 and runes before his path becomes safeish)
Garen (Solotop pubstomper extrodinaire. Very easy to learn. DEMACIA!)
Janna (One of the best - if not THE BEST support int he game.
See this. (Item build section outdated)).
Jax (Jungle/solotop. Kind of meh. Riot never really figured out what to do with this guy. Usually trash but periodically hilariously OP after a remake before nerfs hit again)
Malphite (Jungle/solotop. Decent tank but there are usually better picks).
Morgana (Best AP carry bar none. High skill cap. Always banned in ranked).
Nasus (Pubstomp solotop tanky DPS. Farm-happy but has several hard counters. Dangerous to play in blind pick or firstpicking in draft).
Singed (Requires a special kind of madness to do well with. Highly disruptive tank solotop tank).
Sion Fairly strong, easty-to-learn bruiser. Usually played solotop, solomid or jungle. Farm-happy, high burst).
Taric (Truly outragous support. Pretty decent when paired up with certain carries. Can off-tank).
Teemo (Really, really annoying. Usually solotop. Builds attack speed, AD or AP).
Tristana (Pretty decent but underappreciated AD carry. you can get her for free along with a skin through the facebook promotion).
Tryndamere (Pubstomp solotop/jungle champion. You'll tear people a new one with this guy until you get close to lv30 and they learn how to counter him. He falls off signifigantly when people know how to handle him).
Twisted Fate (Somewhat difficult to learn AP mid. If you stick with him you can pull off some epic ganks, but more likely than not you'll end up feeding. Nerf bat has hurt him hard).
Veigar (Supremely bursty AP carry. Highest burst damage late game bar none, especially against enemy AP carries).
Warwick (solotop/jungle. Babys first jungler #2. Can be jungled even at low summoner levels with noo runes, like nunu)
Zilean (support/ap carry. Pretty decent but not gamebreakingly OP)
TL;DR -- don't bother getting newer champions with IP just yet. For the same IP you'd pay for, say, Fiora, you could get a 10-champion bundle like: Alistar, Tristana, Ashe, Annie, Soraka, Janna, Morgana, Warwick, Master Yi and Poppy