What is the definition of 'belief'... ?

A belief is a proposition that you regard as true.

You're right that we can take on assumptions without regarding them to actually be true. Assumptions are not necessarily beliefs. We can assume things for the sake of an argument, or to see where a line of logic will lead, or to act under uncertainty, etc.

does the act of belief go against reason, that is, is it irrational?

There is no necessary connection between beliefs and rationality. Some beliefs are rational or justified. Others are irrational or unjustified.

What is the definition of 'belief'... ?

A belief is a proposition that you regard as true. (And if justified and true, many would call such beliefs knowledge; but of course not all beliefs are knowledge.)

You're right that we can take on assumptions without regarding them to actually be true. Assumptions are not necessarily beliefs. We can assume things for the sake of an argument, or to see where a line of logic will lead, or to act under uncertainty, etc.

does the act of belief go against reason, that is, is it irrational?

There is no necessary connection between beliefs and rationality. Some beliefs are rational or justified. Others are irrational or unjustified.