What galaxies can you photograph with a DSLR and a 500mm+ telephoto lens?

The largest galaxy (other than the Milky Way) in the sky is that of the Andromeda galaxy and is clearly visible in the night sky of the northern hemisphere with the naked eye - if you know where to look (and realize that fuzzy thing is just lots of stars). The apparent dimensions of the galaxy is 190' x 60' (or about 3° x 1°). To put that in context, the moon is half a degree in radius (it varies between 34.1' and 29.3'). This image shows the relative size of the moon and the Andromeda galaxy.

From What focal length lens do I need for photographing the moon? a reasonable 500mm reflex lens (it has the design of a small telescope - though some have been made that really make you wonder where the telescope or lens boundary is) will work nicely for this. You aren't going to be at all concerned about the bokeh of a reflex lens for astrophotography.

The 500mm lens will give you a field of view of 4° (240') x 2°45' (165'). With this, the Andromeda galaxy should fill the frame.

Exposure is going to be the problem. The photograph described at M31 describes 100 minutes of total exposure to capture it. There are multiple problems that that describes right away. The biggest of these is that the sky moves with respect to where you are sitting. You need the camera to move too. The photograph described used a polar-aligned tracking equatorial mount that was auto-guided.

You are going to want to get a tripod that looks something like this:

(from https://commons.wikimedia.org/wiki/File:Maksutov-Cassegrain_Intes_M703_mounted.jpg)

With the proper setup (including a motor) the tripod's head will track a specific spot on the sky once it is set up. You can read more about the equatorial mount at wikipedia. This design means that the tracking of the sky requires only one degree of motion - it spins on the same axis as the sky does.

Note that this is a key thing. Other mounts such as the altazimuth have multiple degrees of motion required to track the sky. They work if you are looking through it with your eyeball, but they don't track cleanly (they move in steps).

Its possible to make this yourself as a baron door tracker, though it won't be as precise as one that is designed for that sort of work and precision made.

You should also look into broadband light pollution filters if you are anywhere near a city (you should go far away from it if possible given the OP's profile here is a map of light pollution in India and for comparison, a map of light pollution in the United States). One such filter designed for telescope eyepieces: Orion SkyGlow Broadband Light Pollution Filters. If light pollution isn't something you can get away from, consider a Hα filter which gives a narrow range of light which is much less polluted, even in cities (see this example for Andromeda in hydrogen alpha vs composite rgb).

For other objects, the best things to look for would be Messier objects and see which ones interest you. Few will be as big (or 'bright') as the Andromeda galaxy (Messier object #31 aka M31). M31 is the third brightest of the galaxies. Going down a bit, the next brightest galaxy is M33 - the Triangulum Galaxy (which is only about a degree in diameter). Other galaxies (such as M51) are only about 10' or so in diameter and you'd only get hints of their structure with a 500mm lens.