How can we estimate if a certain astronomical object will be visible from a place on earth? If we have the RA/DEC of an object that we want to observe, how can we tell if that object is visible from our place on earth (at a particular time of the year)?

Given that the part of the sky we are observing is changing throughout the year, it seems fairly complicated.
A simple estimate can give you a rough idea what area of the sky will be visible at different times of the year, using only a bit of mental arithmetic.

I just remember one number, the right ascension (RA) at (or nearest to) the meridian (between due S and the zenith) at midnight on Jan 1st. This is (roughly) equivalent to local sidereal time. Actually, I just remember the hour, 06h, at my longitude of just under 72 degrees E. This changes very slowly from year to year due to precession (I may have to use 05h by around 2050 or so, if I am still alive!). Since the difference between sidereal time and UTC is about 4 minutes per day, RA on the meridian at midnight shifts about 2 hours later per month. Now around Jul 15th, about 19h RA is on the meridian at midnight here. Latitude determines what range of declination can be seen. At ~42 degrees N, I usually cannot see much below -30 degrees declination, depending on the horizon.

Knowing what RA is near the meridian (I can easily subtract or add hours corresponding to time before or after midnight), by looking at a sky chart I can easily estimate what objects should be visible, or well placed for observing. I used this simple technique for many years, before digital everything! Now I can just use SkySafari on my iPad. :-)