Finding all possible paths is a hard problem, since there are exponential number of simple paths. Even finding the kth shortest path [or longest path] are NP-Hard.
One possible solution to find all paths [or all paths up to a certain length] from s to t is BFS, without keeping a visited set, or for the weighted version – you might want to use uniform cost search
Note that also in every graph which has cycles [it is not a DAG] there might be infinite number of paths between s to t.