You must allow 5000 Renown to build up, so don't spend it. If you are just getting started best way is to 3 star all of the situations, this will net you a lot of renown quick. The tutorials should net you 800 renown. Next jump into Terrorist Hunt, where you play Co-Op with others. You are most likely going to earn renown faster this way vs playing Vs. with others.
Personally I waited until I had purchased and geared up some, which I believe having better operatives with good gear build outs that work well for your playstyle makes you more successful and thus earn renown more quickly. The above advise still is applicable those are the fastest/easiest ways to move more quickly down the renown path IMHO.
Hope this helps.