I do pretty much what Ironikinit does with factories, except I rarely rush them. Any city that can build a factory within 20 turns is ordered to do so, unless there is something else crucial that needs doing first. In that case, whatever that is gets rushed, so I can start on the factory. The production jump that you get from the combo of RR's and Factories in your core cities is HUGE. By the time the factories are done, I usually have Scientific Method and am working on Electronics (my earlier post put Rep. Parts in front of Electronics, but I realized that I do it the other way around). Also, once I have rep. parts, I usually pump out some infantry. Sometimes it's extra, sometimes it's actually needed to replace Cavalry I've got scattered around defending cities I've taken by conquest during the Middle Ages. Prior to Tanks, I have a lot of cities set to "wealth" which helps me keep the science rate up at 70-90%. Once I have motorized transport, I build a ton of tanks and ponder what to do with them
Hospitals... well, I hold off on sanitation as long as I can (usually get it from Theory of Ev.). Then I grumble about pollution and start building hospitals. But after my cities get up around size 15-16, I send in worker teams to change some irrigation to mines so that they max out at size 18-20. I play on Monarch, and with all luxuries + marketplaces + temples/cathedrals/colloseums + Sistine + Bach, 17 citizens stay happy. I often end up running 10% luxuries so that my few size 20's stay in WLTKD (I like happy people).
-Arrian