What resources are available to find the positions of planets, stars, moons, artificial satellites, asteroids, and other heavenly bodies?
There are many resources online, so I'm making this a community wiki answer. Please feel free to add to it!
If you want to visualize the stars/planets/etc (as viewed from Earth or another location), you are looking for planetarium software: https://en.wikipedia.org/wiki/Planetarium_software
If you want accurate positions for stars/planets/etc, you are looking for HORIZONS:
If you want to compute star/planet positions yourself, you have several options:
SPICE (http://naif.jpl.nasa.gov/naif/tutorials.html) will give you results that match HORIZONS very closely. You can also use some of SPICE's functionality online at http://wgc.jpl.nasa.gov:8080/webgeocalc/#NewCalculation
SPK (Spice kernel) files are available at https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/ -- while these are primarily intended for use with CSPICE, the format is documented and you can use it directly. It's described at https://github.com/skyfielders/python-skyfield/issues/19 and implemented in Python at https://github.com/brandon-rhodes/python-jplephem/tree/master/jplephem
If you want to do the SPICE computations yourself (numerically solving the differential equations), see https://astronomy.stackexchange.com/a/13491/21
You might also want to use an n-body simulator to do the computations yourself: https://physics.stackexchange.com/questions/25241/what-open-source-n-body-codes-are-available-and-what-are-their-features
IAU SOFA (http://www.iausofa.org/) will give you the International Astronomical Union's "official" libraries to compute positions.
VSOP theory (https://en.wikipedia.org/wiki/VSOP_%28planets%29) is another option.
PyEphem (https://en.wikipedia.org/wiki/PyEphem), which derives from MIT's libastro library (https://stuff.mit.edu/afs/athena/project/xephem/src/xephem-3.5.2/libastro/)
If you want to treat planetary orbits as simple ellipses and ignore perturbations, you can find orbital elements at https://ssd.jpl.nasa.gov/txt/p_elem_t1.txt but you'll probably want to visit https://ssd.jpl.nasa.gov/?bodies#elem first
Note that these computational libraries don't always agree with each other or with planetarium software:
If you want a catalog of stars/etc, NOMAD (http://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/nomad) and GAIA (http://gea.esac.esa.int/archive/) are the largest such catalogs (about 1 billion entries each, lots of overlap). The NOMAD link also provides references to smaller catalogs.
For high-resolution astrophotography try ALADIN (http://aladin.u-strasbg.fr/#AladinLite)
If you have pictures of stars you can't identify, try http://astrometry.net/use.html
If you want to find artificial satellites and spacecraft, start at https://en.wikipedia.org/wiki/Two-line_element_set
Sun/moon rise and set times? http://aa.usno.navy.mil/data/docs/RS_OneYear.php
Solar and lunar eclipses? http://eclipse.gsfc.nasa.gov/eclipse.html
Transits of Mercury and Venus across the Sun? http://eclipse.gsfc.nasa.gov/transit/transit.html
Occultations of stars by asteroids? The International Occultation Timing Association (IOTA) at http://occultations.org/
Saw something really bright that didn't look like a regular star or planet? Look over the observation logs at http://www.amsmeteors.org/fireballs/fireball-report/ (if this occurred fairly recently, be sure to look at the pending reports too).
To add to the excellent answer by barrycarter, there are 2 planetarium-like codes, that I know of, that run on a mac and would make excellent tools for viewing certain astronomical objects. The codes are Stellarium and Celestia. Both turn your computer into your own planetarium where you can search and view objects in space.