Systems of time: see Naval observatory reference for a full listing of different types of time!
Time tied to position of Sun. Note the distinction between mean solar time and apparent solar time (the "equation of time" and the analemma).
Most used solar time is Universal time. UT = local mean solar time at Greenwich = UT = Greenwich time = "Zulu". Tied to location of Sun, but averaged to ``mean sun''.
Local time: accounts for longitude of observer. For practically, legal time is split into time zones.
In detail, official time is kept by atomic clocks (International Atomic Time, or TAI), and coordinated UT (UTC) is atomic time with leap seconds added to compensate for changes in earth's rotation, where these are added to keep UTC within a second of solar time (UT1). See here for some details.
Time based on position of stars, i.e. Earth's sidereal rotation period ∼ 23h 56m 4s. Local sidereal time is GMST (Greenwich mean sidereal time) minus longitude. At the vernal equinox (time in sky when Sun crosses the celestial equator as its declination is increasing), sidereal time = UT. Difference between UT and GMST is one rotation (day) over the course of a year, so about 2 hours per month.
Sidereal is relevant for position of stars: stars come back to the same position every sidereal day. As we'll see below, a given star crosses the meridian when the local sidereal time equals the right ascension of the star.
Standard calendar is Gregorian, with leap years, etc. etc.
For astronomy, simpler to keep track of days rather than year/month/day. Most dates given by the Julian date (number of days since UT noon, Monday Jan 1 4713 BC). Variations include modified Julian date (JD - 2400000.5, fewer digits and starts at midnight), heliocentric Julian date (JD adjusted to the frame of reference of the Sun, so can differ by up to 8.3 minutes).
Note that repeating events are often described an an event ephemeris: ti(event) = t0 + i(period ).
The term ephemeris is also used to describe how the position of an object changes over time, e.g. planetary ephemerides.
LPL website on astronomical coordinate systems
At vernal equinox, RA=12h crosses the meridian at midnight.
Note that for a celestial coordinate system tied to the Earth's rotation, coordinates of an object change over time because of the changing direction of the earth's axis: precession and notation. Because of this, coordinates are always specified for some reference equinox: J2000/FK5, B1950, ect.; if using coordinates to point a telescope, you need to account for this (but generally, telescope software does!). Note distinction between equinox and epoch, where the latter is relevant for objects that move (which everything does at some level!).
Transformations between systems straightforward from spherical trigonmetry.
Note the common usage of an Aitoff projection (equal areas) of the sky in celestial coordinates, with location of ecliptic and galactic plane. Software tools (Python, projection=``aitoff" in subplot, IDL: aitoff and aitoff_grid in Astronomy users library).
Local coordinates are important for pointing telescopes! Note that there are various other effects that one has to consider for pointing a telescope at a source of known celestial position: proper motion, precession, nutation, ``aberration of light'', parallax, atmospheric refraction.
Usually specified by position angle: angle of object in degrees from NS line, measured counterclockwise.
An important observational position angle for spectroscopy: parallactic angle, the position angle of the line from zenith to horizon
Obviously, to observe an object, one requires that it is visible above the horizon. In general, one would like to observe objects through the shortest possible path through the Earth's atmosphere, i.e., when they are transiting (crossing the meridian, HA=0). The more atmosphere the light goes through, the more losses due to atmospheric absorption/scattering (more severe at shorter wavelengths), and the more image degradation from atmospheric seeing. Of course, it doesn't make sense to wait for an object to transit if you don't have anything else to do in the meantime; efficient use of telescope time is primary concern. Generally, most observers attempt to observe at airmasses (ask if you don't know what this means!) less than 2, i.e. within 60 degrees of the zenith. Once you hit an airmass of 3, the object is rapidly setting (except at very high declination). Of course, for some solar system objects (objects near the Sun), one has no choice but to observe at high airmass.
Note that HA gives some indication of observability, but that higher declination objects can be observed to higher HA than lower declination objects. Roughly, at the celestial equator, an HA of 3 hrs is about an airmass of 2, and in many cases, one doesn't want to go much lower in the sky.
Another issue with observability has to do with the Moon, since it is harder to see fainter objects when the sky is brighter (do you understand in detail why?). Moon brightness is related to its phase, and to a lesser extent, to distance from your object. Of course, if the Moon is below the horizon, it does not have an effect. So for planning observations of faint objects, one also has to consider Moon phase and rise/set times. Note that the sky brightness from the Moon is a function of wavelength, and at IR wavelengths, it is not a very signficant contributor to the total sky brightness: so often, telescopes spend bright time working in the IR.
Here are some useful software tools to do tasks related to coordinate systems and observability. There are certainly other tools out there. Anything that accomplishes the desired tasks adequately is fine to use! Just make sure you're not limited by the tools that you choose. These are available on the Astronomy Linux cluster; you can probably install them on your laptop, but they will probably not be there by default.