A galaxy is what you get when gravity gathers a few hundred billion stars into one turning system and keeps them there. There are about two trillion of them. You can find one of them tonight with no equipment at all.
Stars are not scattered evenly through the universe. They come in enormous clumps, held together by gravity, with almost entirely empty space between the clumps. One clump is a galaxy. Ours is called the Milky Way, and it is an ordinary one — big, but nothing special.
They come in a handful of shapes, and the shape tells you the history. Spirals are still making new stars in their arms. Ellipticals have run out and are coasting on the old ones. Irregulars have usually been pulled out of shape by a neighbour — and some have been through an actual collision.
The Sagittarius Dwarf Galaxy and Streamdwarf galaxyNot observable
Andromeda Galaxy (M31)spiral galaxyNaked eyeTriangulum Galaxy (M33)spiral galaxyWith effortLarge Magellanic Cloudirregular galaxyNaked eyeBode's Galaxy (M81)spiral galaxyWith effortWhirlpool Galaxy (M51)spiral galaxyWith effort
Sombrero Galaxy (M104)spiral galaxyWith effortPinwheel Galaxy (M101)spiral galaxyWith effort
Centaurus A (NGC 5128)elliptical galaxyWith effortM87 / Virgo Aelliptical galaxyWith effortCigar Galaxy (M82)starburst galaxyWith effortThe Antennae Galaxies (NGC 4038/4039)colliding galaxiesNot observableCartwheel Galaxyring galaxyNot observableHoag's Objectring galaxyNot observableMalin 1spiral galaxyNot observableAndromeda is the far end of your own eyesight: on a decent night, in autumn, it is a faint smudge you can find without any equipment at all. That smudge is two and a half million light-years away, which means the light entering your eye left before there were people.
Expect a smudge. Every galaxy photo you have ever seen took hours of exposure on a telescope the size of a house. Through a small one, a galaxy is a pale grey oval with no colour and no visible arms. Knowing that in advance is the difference between a good first look and a disappointing one.
Far enough that the question was genuinely open until about a hundred years ago — people argued about whether these smudges were inside our own galaxy or were galaxies themselves. The answer came from a particular kind of star that pulses at a rate which gives away exactly how bright it really is.
Autumn, a moonless night, a clear northern horizon, and Andromeda. That is the whole assignment, and it is enough.