Without a save game I can't tell you what mistakes you're making. I can tell you what mistakes I made at first on noble.
1.Do you really really really like wonders? I love wonders. They're wonderful. The problem is, sometimes you can make them too much of a priority and neglect your military. Never neglect your military. Wonders go from wonderful to worthless without one.
2. Plan your National Wonders accordingly. If you have a holy city or found a corporation this is where you want your stock exchange. I usually build heroic epic in my 2nd or 3rd most productive city not my 1st my 1st is for wonders. Don't make this city the greatest place to live, make it a functional city whose job it is to make ass-kickers. Build Iron Works and National Park in the same city.
3. Keep an eye on your neihbors' religion. If they are near you and powerful, try to convert them (go for their capitol first). Sometimes you may need to convert to theirs, even if it's a pain. It helps diplomacy very much.
4. Keep an eye on how powerful the other civs are and who likes who. You won't be able to please everybody (most likely). Pick the people who trade techs (even if they're unfair trades, some civs don't seem to trade at all) to keep as allies. Decide who's worth appeasing and who is not.
5.Pay attention to specialists, free specialists, and wonders' great person impact. Example - You built the oracle, chitchen itza and your shrine all in the same city. You produce a lot of great people in this city. You could produce more with a national epic. Except.... a lot of them will be priests. Which gets old half way through the game and gets older later in the game.
6. Check out these forums. Upload your saves a lot of people will give you precise advice. A lot of people disagree. Smart people can disagree. But you'll see this forum is like Rome's Unique Building it increases great person output.
7. Expansion is expensive. Courthouses are key, much more key than market places if you ask me. Which brings me to point 8.
8. You don't need every improvement in every city. If you have a holy city w/ shrine, and you build a market, grocer, bank, harbor, customs house, airport on that city it will make up for 7-8 cities without them. A lot of hammer cities don't make much use out of marketplaces.
9. Check the tech trading page every turn. Re-evaluate resource trades every 30.
10. Don't give up I think it took me 6 tries. Don't go back to warlord. Noble is what separates the zeros from the heroes. If I'm ever talking to someone in real life about civ and they don't play on noble or above I pretty much lose interest. No offense to anyone who plays warlord. Nothing's wrong that, it's just if you don't ever plan to play on noble, I can't take your advice too seriously. Which maybe is how people who play on Emperor or above think about me. Oh well.