Where can I find the positions of the planets, stars, moons, artificial satellites, etc. and visualize them?
There are many resources online, so this is 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:


https://en.wikipedia.org/wiki/JPL_Horizons_On-Line_Ephemeris_System
http://ssd.jpl.nasa.gov/?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 &quot;official&quot; 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/)

Skyfield: https://github.com/skyfielders/python-skyfield

libnova (http://libnova.sourceforge.net/)

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:
https://stackoverflow.com/questions/16293146

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 want to find artificial satellites and spacecraft, start at
https://en.wikipedia.org/wiki/Two-line_element_set

Transits of Mercury and Venus across the Sun? http://eclipse.gsfc.nasa.gov/transit/transit.html

Planetary conjunctions?
http://laps.noaa.gov/albers/ast/conjun/conjun.html

Occultations of stars by asteroids? The International Occultation
Timing Association (IOTA) at http://occultations.org/

Create your own Sun-Earth-Moon model by using a ~17 inches (~44 cm) beach ball or box as the Sun, 4-5 mm object like a pencil eraser as the Earth and a 1 mm sized drill bit or other object to represent the Moon. Then, find a 150-foot string or tape measure. The sun and earth are ~154 feet (~46m) apart. The earth and moon are ~4.7 inches (12 cm) apart.