By Jennifer Agiesta, CNN Polling Director

(CNN) -- As Congress turns its attention away from health care and back toward tax reform and other issues, a new CNN poll conducted by SSRS finds most in favor of major changes to the nation's tax laws, but wary of President Donald Trump's handling of immigration.

Tax reform and immigration have both been near the top of Trump's agenda, but the public sees them as lower priorities. Just 12% polled said tax laws ought to be Congress's highest priority over the next few weeks, 11% choose immigration. More, 36%, want Congress to focus on funding disaster relief efforts (the poll was completed September 17 through 20, before the effects of Hurricane Maria on Puerto Rico were clear) and 31% say dealing with health care should be the top priority.

Neither taxes nor immigration prompts much support for Trump, according to the poll. Overall, just 36% approve of his handling of immigration -- down eight points from a high of 44% in March -- and 34% approve of his work on taxes. A sizable one in five say they're not sure yet how Trump is doing on taxes, as details of the tax reform effort he plans to advocate for yet to be released.

Many Americans agree with Trump's contention that the tax laws deserve a major revamp. All told, 68% say the federal income tax system needs either a complete overhaul (35%) or major changes (33%). That cuts across party lines, including 77% of Republicans, 70% of independents and 62% of Democrats.

Asked about the impact of a reform effort led by Trump, though, about four in 10 say they expect taxes for the middle class to rise, while just 25% say middle class taxes will drop. Likewise, more Americans think their own personal taxes would go up under this effort than drop (34% to 21%). More see tax decreases on the horizon for big businesses (47% say so) and for the wealthy (42%).

These views are divided by party. Republicans are more likely than Democrats to say their own taxes would drop under a Trump-led plan (40% among Republicans, 11% among Democrats) and that middle-class taxes would decline (52% among Republicans, 9% among Democrats). But Democrats are more convinced that the wealthy would see a decline in their tax bills (61% vs. 19% of Republicans) and that big businesses would also get a break (57% vs. 36% among Republicans).

Turning to immigration, most (60%) lack confidence that Trump and Congress will be able to enact new laws on immigration and border security that will improve the way the country handles this issue. That's similar to the sentiment toward then-president George W. Bush's efforts at immigration reform in mid-2006.

That lack of confidence could stem from disagreements on policy. The poll finds most Americans oppose each of the major initiatives Trump has backed on immigration since taking office: 82% say they want the policy known as DACA to continue, 63% oppose building a wall along the entire border with Mexico, and 55% say they oppose changes in federal law to reduce the number of immigrants who enter the country legally.

In dealing with those already in the country illegally, a whopping 84% say developing a plan to allow some people living in the US illegally to stay is a more important priority than deporting all people who are living in the US illegally. Considering how to deal with those attempting to move to the US, most favor reforming current laws to better reflect the country's needs than say stronger enforcement of current laws should be the top priority (56% to 40% for enforcement).

And in principle, many seem to support at least one of the changes Trump has backed to immigration laws: 51% say they think job skills and education should be a more important factor than family ties in considering whether to grant a non-citizen legal residence in the US But those who see reforming immigration laws as a priority are split on this question, with 46% favoring job skills and 43% family ties, and opposed to reduced legal immigration generally (65% oppose it, 30% favor it).

The CNN poll was conducted by SSRS by telephone September 17 to 20 among a random national sample of 1,053 adults. The margin of sampling error for results among the full sample is plus or minus 3.7 percentage points; it is larger for subgroups.