Which party is trusted more on the issues? It turns out that depends on which issue you are talking about. At least according to this survey done by the Pew Research Center between April 5th and 11th of a random national sample of 1,501 adults (with a 2.9 point margin of error).
The public trusts the Republicans more on terrorism (by 12 points), gun policy (5 points), economy (3 points), and trade (3 points).
They trust the Democrats more on government spending (8 points), immigration (11 points), foreign policy (13 points), education (16 points), health care (19 points), abortion (20 points), and environment (31 points).
The public is split on who they trust more on taxes (Republicans 44% to Democrats 43%). That one point difference is well within the survey's margin of error.
This split probably helps to explain why the public has such a poor view of Congress. They simply don't trust either party on all the issues.