Sid Meier's Colonization/Founding Fathers

Bells of Liberty are extremely useful. They make your cities a lot more productive, get founding fathers into your Congress, and improve your bonuses in the War on Independence (they will get you a few free units in the war too).

Fathers
The fathers are (pluses denote importance): When finding treasure from Lost City rumors, transport becomes very handy. Send out a couple of scouts in various directions, and they'll soon pay for themselves.
 * Adam Smith (+++) - lets you build factories, very useful in middle and late game; factories 50% more output than their input (so you put in 20 ore and get 30 tools); Iron Works is the most important such building
 * Jakob Fugger (++) - cancels all boycotts, a very useful but one-time effect
 * Peter Minuit (+++) - all Indian lands are free, get as soon as possible; You can probably play without it (in Spanish "screw all Indians" strategy Cortes would be more useful), but it doesn't seem like a particularly good idea. Indians want too much money for their land and taking as little as 2 squares for free is enough to make them attack you. So take only no-Minuit strategy I can think of is an early war against the Indians.
 * Peter Stuyvesant (++) - lets you build Custom House, important but not urgent
 * Jan De Witt (+) - trade with foreign colonies becomes possible and foreign reports are become more detailed. I usually don't trade with others, but it seems to be a viable special case strategy.
 * Ferdinand Magellan (+) - all vessels get +1 free movement point, travel time from western map edge to Europe gets shorter
 * Francisco de Coronado (-) - all colonies become visible; completely useless (some Readme says "no more Indian ambush [bonus ?]" with Coronado, but I haven't verified that)
 * Hernando De Soto (++) - results of exploring Lost City Rumors are always positive and all units have extended sighting radius; It means at least twice as many Galleons full of treasures and fountains of youth
 * Henry Hudson (++) - doubles output by the fur trappers; useful, but not as much as it seems because the fur prices fall due to increased supply
 * Sieur de La Salle (+) - all colonies of size 3 get free Stockade; useful, but it also means you can't get rid of such colonies. You will need a Stockade in each colony, so with 10 colonies that's 640 free hammers. It's not much.
 * Herman Cortes (++) - conquered native villages get you more treasure and the king transports it for you for free; free transport is not very useful - when you get to the stage when big wars with Indians are sensible, you already have at least 1 Galleon. Bigger treasure, on the other hand, is great.
 * George Washington (++)    - all soldiers become Veterans after a won fight, useful in middle and late game.
 * Paul Revere (-) - if a colony with no army but stockpiled guns is attacked, colonists take the guns to defend themselves; probably the most useless
 * Francis Drake (++) - all your privateers get +50% strength
 * John Paul Jones (+) - you get a free frigate. That's only 512 free hammers and 200 free tools, and spent on something that's not very useful. Building or buying a Frigate may be a better idea (if you actually need one, as you probably don't).
 * Thomas Jefferson (++++) - your liberty bells production increases 50%; very important and urgent
 * Pocahontas (++) - tension level between you and the natives decreases to normal and alarm is generated half as fast; having French and Pocahontas makes the Indians calm even when their villages are in center of population 100+ empire
 * Thomas Paine (+) - liberty bells production is increased by the current tax rate; may be useful in late game, depending on the tax rate, but Bolivar and Jefferson are much better
 * Simon Bolivar (++++) - sons of liberty membership increases by 20% in all colonies; very important and urgent
 * Benjamin Franklin (+) - all other powers always offer peace in negotiations; they do it most of the time even without Franklin, as long as you agree to get your armies and privateers out of their colonies.
 * William Brewster (+++)    - no more criminals or servants in the docks and you get to select 1 of 3 available immigrants; it's probably not a good idea, but you can play without Brewster; any number of servants can be upgraded to specialists and then downgraded to free colonists in any Indian capital, criminals can work as missionaries, especially after Brebeuf (but Brewster is usually much more useful than Brebeuf), or you can educate them to servants and then send them to Indians
 * William Penn (+) - production of crosses increases by 50%
 * Father Jean de Brebeuf (+) - all missionaries function as experts; so you can turn useless criminals into about as useless converts.
 * Juan de Sepulveda (+) - increases chances that Indian convert after their villages are attacked
 * Bartolome de las Casas (++)- turns all Indian converts into free colonists, a very useful but one-time effect

Strategy
Get Minuit and Brewster immediately. Time one-time effects (Fugger, Casas, Pocahontas has one-time effect, but her future effect is more important) very carefully. If you don't know whom to select, get those with more pluses on this list.

Washington, Bolivar, Fugger, Jones and Casas can only appear after 1600.

In some versions of the game if you press escape instead of choosing one of the fathers you will get a different selection next turn. That's borderline cheating, don't do that.

Each father needs more and more bells. It's nothing like Civilization's "invention every 2 turns". Cost of fathers on lower levels is significantly lower than on higher levels, and each father costs more than the previous one, so if you play Conqueror you can actually take every useful father (Drake, La Salle, Penn) without much thinking, but on Governor/Viceroy you must be more discriminating.

You need a lot of bells for colony efficiency, so you will eventually get all the important fathers, but probably not all until very late game. Detailed calculations in the next section.

Summary of the table

 * As soon as you can: Jefferson, Bolivar (only after 1600)
 * Then, unless you have some unusual strategy: Minuit, Brewster
 * Extremely useful: Smith, Stuyvesant, Washington (only after 1600)
 * Very useful, but depends on your strategy: Fugger, Soto, Hudson, Drake, Casas, Pocahontas, Cortes

If you want peace with the Indians
Basically, if you're going for peace with the Indians, prioritize:
 * Minuit (land)
 * Brebeuf (missionaries)
 * Pocahontas (long-term peace)

If you are going for conquest
If you're going for conquest, then get:
 * Cortez (gold)
 * Sepulveda (converts)
 * Las Casas (after you've gotten as many converts as you can)
 * Pocahontas (to secure trading partners after the immediate pillaging is over)

As you may notice, some founding fathers are always useful, while others are very dependent on strategy. It's important to select a general strategy early, so you can focus your selection of the FF effectively.