First of all all of you questions depend on your personal preference. But I'll come to a conclusion after answering your questions.
Should I spend my IP only on runes or as well on champions?
This depends on what champions you like to play. You can get up to lvl 30 in League of Legends before the game considers you maxed level. At level 20 you unlock the best possible runes, marks, glyphs, and seals. As you can see the earlier runes give you a bonus but it's usually not worth getting them since you're going to replace them after a few days of playing anyway. The thing to probably note as a "new" player is that you dont need to buy runes before 20. The difference those runes make is neglect-able and not worth compared to their cost.
"So what do i do with all of the IP i earn then?"
Well that's a good question. Before you hit level 20 you obviously earn a lot of IP simply by winning and losing matches. Now how much you spend of it is entirely up to you. You could decide you only want 1 champion and after that you save for the runes you're going to buy once you hit level 20. That's a possibility. This is however not the most exciting way of playing the game. (once again its all personal preference, but) personally i prefer(ed) to buy the champions i 'liked' to play until level 18 or so (17, 18, 19) and then save your IP for when you hit lvl 20 to buy runes. More about this on your other questions.
When does it make sense to start buying RP?
League of Legends is a free to play game. You can get to the same level if either pay or don't. I can try to explain this but the league of legends wiki has the perfect explanation already:
They can be purchased with real money and are used to purchase in-game things to add convenience, extra customization options, or diversity to the experience. Examples of things that can be purchased with Riot Points are champions, alternate champion skins, or boosts, which are modifiers that increase the rate at which you gain Influence Points or Summoner Experience Points. Apart from purchasing champions and extra rune pages, which are both also purchasable with Influence Points, Riot Points cannot be used to directly affect gameplay.
Especially note the last line "Riot Points cannot be used to directly affect gameplay." Meaning that you have the exact same cooldowns, costs, movement speed etc as all other summoners in the game (with the exception of the runes).
What should I buy with RP, champions or boosts?
Once again what you decide to buy with your RP is entirely up to yourself. Buying champions with RP is only if you don't want to go through the trouble of getting the IP since every champion is unlocked with IP as well as RP. The thing i use RP for are skins. I now own nearly every skin in the game and i enjoy playing with the skins as well. You could also buy boosters but this only increases either your IP or EXP.
Should I only buy champions I already played on a free week?
Buying a champion which you've already played does give you an edge in playing that champ but most importantly it ensures you 'like' to play that champion. For example i bought the champion Lucian right when it got released but i didn't like to play with him. Should you ONLY buy champions you've already played? No, but its up to you.
When should I start buying runes or champions (which level)?
At level 20 (see your first question).
If you have any other questions, feel free to ask.