You must try and get the Great Library wonder it can make a huge difference but hold off researching education - as it will negate the Great Library.
On chieftain the Great Library will be rather useless.
Four: American stink... choose a civ with a more scientific profile and a civ with a specific unit that come before the modern age. That should help you get a better start...
You don't need a UU to get a golden age. Industrious is rather useful. Last game playing the French on Monarch level I got my golden age from the Colossus. Won by spaceship.
Third: War War War War War... and democracy... war is a great way to expend quickly even if you do not want a military victory and if you don't want to be savage, at least take a good governement... no communism for the peace like...
War is not necessary, but can help. By claiming more land you'll get more cities, and with war, you might get a great leader that can be used to rush the Forbidden Palace for example. It would be smarter just to stay in Republic than to keep going in and out of anarchy when switching from Democracy to a war-time government.
Six: in your major cities and most importantly in those with the wonders I talk above, place scientist. You can change them in the city view by removing citizen from the working squares, that gives you an entertainer that you'll need to change to a scientist by clicking on them. That should improve your city science's production.
NO, NO, NO!!! Not in your major cities. In your low-corrupt cities you need as many citizens working the tiles as possible, because all of the shields/commerce they bring in is uncorrupted, plus the libraries, universities multiply this gold. Scientists add just 1 beaker! In the high (95%) corrupt cities, then scientists would be good, because the beakers are corruption immune.
I have not played chieftan in quite a while and can't remember much about it. Is this another surprise from Firaxis? In the editor it has a tech rate in each difficulty level, but that is supposed to be for AI only.
Difficulty does not slow down the human's potential to research. The reason it seems that on lower levels it takes so much longer to go through the tech tree is because:
1. On chieftain, after you get the first couple of techs that the other civs start with or techs they got from goody huts, you pretty much have to research all of them yourself. On higher levels you can trade quite a bit for other techs that you didn't research yet alot further into the tech tree. Trading industrial age techs for example happens alot more on the higher levels than chieftain. In the industrail age, it takes the AI 40 turns for every tech it seems. It takes the computer so much longer to build infrastructure that they do not have the resources to research those expensive techs.
2. On lower levels, due to poor infrastructure, the AI has no money to give the player to help/trade the player to speed up his research.
My suggestions: Like has been said before, make sure you have roads everywhere. click on a city and zoom in, see which tiles are being worked. Make sure those tiles have roads on them (don't completely ignore mines/irrigation though). Build Libraries, courthouses, universities, etc. Make sure your people are happy, entertainers won't help you at all in production or research, so secure some luxuries and build happiness improvements. I recommend being a Republic if you want to take the scientific route. Ignore the dead end techs unless you really need it for a specific wonder/military unit, etc. Let the AI research those techs for you and buy it from them. Or if you are at the same speed technologically as the AI is, researching those dead-end techs might be smart because you could trade techs.
On chieftain I recommend having the science bar set to 70%, or the highest you can afford and still make a little gold each turn. Try to get Copernicus's Observatory and Newton's University in the same city. In the last game I played I had both those wonders, plus colossus in my capital and when I set science to max, I was making over 200 beakers/turn from just that one city. Later on, I got SETI program in that city also, but the spaceship was launched just a few turns later.