It is often quoted, including on this site, that space has nearly infinite resources to offer.  Of course, if they were all conveniently and obviously located, we would be using them now, and this discussion would be totally unnecessary.  So here, I will discuss the resources we can find in various places in the solar system and a little of how they might be used.

The first resource many people look for in space is either breathable air or rocket fuel.  Since oxygen is the primary constituent of both of these, location of an extraterrestrial source of oxygen is of primary importance.  The best source of oxygen is water, since it also fills a life-support need, and oxygen and hydrogen mix to become the best chemical rocket fuel available.  Until a few years ago, however, space was thought to be a very dry place.  Recent discoveries, however, have suggested that there is a small amount of water on the Moon, and information supplied by the Gallileo probe to Jupiter indicate that there may be significant water supplies on three of Jupiter's moons.  A second source of oxygen alone is the surface of the Moon, which is approximately 40% oxygen by weight.  There are some difficulties with the liberation of the oxygen from the mostly Iron and Titanium that it is bound to, such as the need to import Hydrogen for the process, but it does provide copious quantities of Oxygen, along with the useful construction materials Iron and Titanium.

Rocket fuel is an interesting question, since there are a variety of options.  For a chemical engine, the fuel has two components, the actual fuel and oxygen to burn it with.  The fuel can be a variety of substances.  For efficiency, the best fuel is hydrogen.  However, hydrogen has several drawbacks.  One, it can be difficult to obtain:  the Moon and Mars, for example, are chronically deficient in hydrogen.  Second, it is very bulky, and the increased size (and therefore mass) of the tanks can take away the advantages of efficiency.  Third, it is a very low temperature cryogen (as a liquid), so it is extremely difficult to handle (many materials cannot take the extreme cold), and it needs active refrigeration, even in space, to keep it from boiling off.  Other fuel choices are methane, which could be produced on Mars with the importation of a small amount of hydrogen, and a variety of other substances.  In the case of a thermal (solar or nuclear) or electric-ion engine, only one fuel is needed, as it does not have to chemically react.  For this type of engine, almost any substance that has good working properties will do, although the lighter elements are more efficient (leading to hydrogen again, if possible).  As was discussed above, there are possible water supplies that have been located, which can provide hydrogen in addition to oxygen.  Many asteroids are also believed to be rich in hydrogen, and comets are believed to have extensive water ice.

Power is an obvious necessity, and may be the easiest to obtain.  Solar power is relatively cheap and is available anywhere in the inner solar system.  Unfortunately, however, it isn't very dense, so it requires very large collectors to generate large amounts of power.  Conventional nuclear power (fission) seems to only be an option if we launch the reactor fuel from Earth, which might be acceptable for limited use on small probes, but is not likely to be allowed (or should be) for something on the order of engines for major spacecraft.  To my knowledge, no significant uranium deposits have been located on extraterrestrial bodies, although I do not believe we have searched in depth.  In contrast, fusion may provide an enormous power source in space, as hydrogen-2 and -3 and helium-3 can be mined from the Moon (or any other airless body), Jupiter's upper atmosphere, or even space itself (the solar wind).  In the case of surface bases, geothermal power might be an option (assuming the body has a significantly warm inner core).  While more advanced options may become available, and will be necessary for colonization beyond this solar system, these options should be more than sufficient to maintain life at or above current standards for the foreseeable future anywhere in this solar system.  Note--anti-matter has not been discussed here because it is not found naturally except on the event horizon of a black hole (or other such exotic place), but must be created by one of the means above...it is more a storage system than an energy source of its own.

Having power, rocket fuel, and oxygen available, the next priority is building materials.  Thankfully, metals are much more abundant and easily obtained in space than on Earth.  The Moon has large quantities of iron and titanium on its surface, as discussed above, the refining of which provides oxygen.  Aluminum is also common, along with silicon (for computer chips and glass), as is large quantities of dirt and rock to be used for shielding and such.  The various asteroids have large quantities of various metals, most importantly iron and nickel, along with larger shares of "strategic minerals" than are found in Earth ores (platinum, gold, cobalt, etc).  Mars, the Moon, and several other large bodies, having been formed through much the same process as Earth, should also have significant localized concentrations of metallic ores buried beneath the surface, although no serious search for such resources has been attempted.

Finally, to sustain life 100% without Earth resources, a suitable supply of light elements and organic matter must be found.  It is believed that the asteroids and certain large bodies (such as the Saturn moon Titan), are home to significant (asteroids) to large (Triton) quantities of such materials.