How can I earn new champions faster?

On League of Legends, you can buy champions through use of RP (Purchased through real-world cash) or IP. You don't need to spend a fortune since IP is earned through just playing. If you queue up with more players, you can get more IP per match. You still get IP, albeit less, if you lose.

Riot (The developers) is actually testing a new party reward program where if you queue with more friends, you get more IP per match. They also have these events occasionally, so keep a lookout. However, this feature may not be released for a while.

You can also buy IP Boosts for RP to temporarily increase the amount of IP you gain per match.

IP is actually gained linearly throughout a match's duration depending on how many minutes you play for. The minutes are always rounded down regardless of how many seconds. (A 39:52 minute game is 39 minutes worth of IP.) Also, Co-op VS AI modes always earn a lot less IP, so be weary of that.

Map                 Summoner's Rift    Twisted Treeline    Dominion
[IP / min] (win)       2.312 IP           2.312 IP            2 IP
[IP / min] (loss)      1.405 IP           1.405 IP            1 IP

Just save up as much as you can to buy a champion you want, and have fun while playing. Also, defance's advice is pretty good.

As mentioned in the other answer, IP is going to be the normal method of buying new champions. There are several ways to increase your IP gains - most notably, you can purchase an IP boost (which will increase the IP you gain per victory or for a certain time frame) using RP (Riot Points). There is also the First Win of the Day bonus, which will grant you 150 IP just for winning a game - and bots games do count for this bonus. More details about IP gain can be found here.

You can also just buy champions directly with RP. RP is bought with real-world money, so this is perhaps a waste... but if you're really certain you want a champion and you don't want to wait, this is certainly a valid option.

Finally, you can also earn champions from the Hextech Crafting system. Whenever you earn an S- ranking or higher on a champion you own and haven't previously earned one on that champion this season, you will gain a Hextech chest. These chests can be opened with Hextech keys, which are are earned randomly whenever you win a game. When you open the chest, you have a chance to gain a "champion shard". If you earn three champion shards, you can reroll them into a random champion that you don't own. Alternatively, if you have enough champion essence, you can directly unlock one of the champion shards you've collected (so that you know what champion you'll get).