It depends on how much you play of course. If you play custom games (essentially where it is just you and bots, even on your team) you really don't get a lot, but if you play Co-op vs AI (bots) in matchmaking, as in you create a party or roll solo and let them match you with a team, you get a lot more. ARAMs is a pretty quick way to get some levels too. Probably looking at a few months if you play at a regular pace. Especially if you get sucked into it like most.
There aren't really counters to support and marksman; they are positions, archetypes if you will, on the map. Generally you will lane against an opposing pair of marksman/support (there are some advanced tactics like swapping solo top to bottom that could be consider countering in all technicality but you shouldn't worry about that yet). Individual champions can be countered, but when you're just getting into the game, you are much better off just finding a small champion pool that you feel comfortable with and enjoy playing than trying to counter your opponent.
And yes, Riot makes the money. When new champions release they are generally more expensive and decrease gradually over time, sometimes go on sale. Also, there is are two different currencies: RP and IP. RP you buy with real money, and a new champion generally costs 975RP ($10.00 for all intents and purposes, which is actually 1380RP but the smallest denomination you can buy and have enough to buy the champ). IP on the other hand is earned from playing the game and so naturally the cost is a bit higher: 6300IP for example for a new champ.
But definitely not 7800RP... that would be over $50 for a champion. They haven't spiraled that much out of control... yet.