I feel like these cities are just so-so, since they have their food in the second ring (the fourth one doesn't even have food). Will this be enough for the rest of the game? How many cities should I have at each point in the game?
Also would I be building a monument first in these cities?
Don't settle a spot with no food. Food in the 2nd ring is ok because at some point you will run caste system and just put an artist specialist for a few turns to grow the borders. Monument is not necessary unless you need the extra happiness (because Washington is charismatic, non-CHA leaders don't get this benefit from monuments).
For a science game you need at least 6 good production cities. Generally these are cities with a lot of green land that you can convert into workshops and watermills in preparation for state property. Those cities will have a forge and one, usually the best production outside the capital, will get the Ironworks. They will also have a levee (if on river), a factory with power plant, the health buildings, an observatory (prereq for the lab) and a laboratory (to speed up spaceship part construction). The capital is can usually hold a few production buildings, including the forge, even if it stays mainly a commerce/science city. You generally want Oxford university in the bureaucracy capital, and that requires 6 universities in the empire. Those can be hard to build in a non-production-rich city due to the high cost, but you can plan ahead and save some forests to chop.
It's a good idea to have closer to 20 cities for a faster space date, so you want to grab an AI's best cities to supplement your own and cover them in workshops/watermills.
How do I decide what is best for the great people? Is an academy bad? Why was lightbulbing Philosophy a bad play (can great scientists even choose what techs to lightbulb?)
See here for predicting what great people will bulb:
Edit: My original article (at bottom) is obsolete after lots of community research into GP tech preferences. Thanks to notagoodname from the civ4 multiplayer ladder site for putting this new article together.
In the game great people have a list of techs and a preference for each tech (no techs have the same preference). With the same techs researched the same great person type will always give you the same tech.
First: Here's complete list of techs a prophet will get in the order they will try and get them (preference of tech they give you runs from top to bottom)
Great...
Also, it looks like you're playing with the BUG mod, and it shows the bulbs on the left side of the tech tree screen. Next to each great person icon, the first tech icon shows what it would bulb now, and the 2nd shows what it would bulb if you research every tech you have queued up.
For a science victory plan, an academy in the capital is a must and you might has well use your first great scientist for that (since then you get its effect for the longest time possible). The philosophy bulb is good otherwise because it is on the path to liberalism and unlocks a civic that accelerates the next great people (pacifism, if you use it).
Do I just let my cities grow into unhappiness/unhealthiness until it's time to build something? My perfectionist side feels kind of weird about that. Wouldn't that be a waste of population?
It's correct, if you are waiting to unlock something you want to build right away, be it military or infrastructure. However, in a science victory game (unlike other victories) you are in for the long run and will build a lot of buildings eventually, so it's good to get a headstart on that, as long as you build in the right order. The granary is usually the first building in most cities to speed up growth, then maybe a lighthouse if it's coast. Any future production city (e.g. a city to be covered in workshops / watermills for state property) should build the forge soon after that if available, because then it makes all other builds cheaper. It also helps with happiness problems. Harbor is cheap for Washington due to expansive trait, but generally the health buildings (Grocer / Harbor / Aqueduct) wait until after factories when health really starts to be an issue.
Also another question I had: I should probably get into trading techs right? I haven't been doing that (or any diplomatic stuff at all for that matter) this game. Which techs should I prioritize trading? (I kind of answered this question myself, elizabeth and alexander are both willing to trade hunting I guess)
Yes, you should trade for techs you need, and if the AIs have accumulated some gold (check the foreign advisor under techs), you can sell old techs to them to accumulate gold to run the slider at 100% research longer. Just don't trade your latest techs or some that will unlock units you don't want your future war target to have. If you trade a tech for gold, it's better to trade it to other AIs on the same turn or the next one, since otherwise they might trade it with each other and you lose potential benefit.
Trading to the AIs an old tech for cheap, or just giving it to them, also is one of the easiest way to build positive relationships (it will display up to +4 "Our trade relations have been fair", and by 'fair' they really means strongly in their favor).
. It does tend to break even within 40 or so turns, but as it shifts the benefits to a later time, it actually needs to save a lot more gold to be worth it. Thus it typically slows down key techs at first and then barely speeds them up, making them a bad choice.
for financial leaders, or 8 with free speech.If you are going for space your good cities will want libraries and observatories, with only a few building universities. Labs do not count as they come too late to pay back on research, they are only built for the production modifier. This results in a 50% modifier for science, or a town being 9 
, resulting in 8.4...
