For example, the word かばん (bag) is pronounced kabaN when it's alone but what if another word like と (and) comes after it? Would it be still pronounced kabaNto, or should you consider the following と and say kabanto?
Yes, it does. For example, ん is pronounced like [n] in かばんと、かばんの、and かばんで (before [t], [n] or [d]), like [ng]/[ŋ] (as in siNGer) in かばんが and かばんか (before [g] or [k]), and like [m] in かばんも (before [m], [p], or [b]).
Adding to the other great answers, with かばん, the N, for linguists, is a uvular nasal /N/, almost like a real nasal vowel.
The original /N/ pronunciation is also retained, or become a nasal vowel before approximants (like y and w-sounds: や、ゆ、よ、わ...）as in ほんを、こんや、...
In general, the ん sound can also vary depending on the speaker. Although, all this stuff is what I've learned so far. Please correct my errors too!