New housing starts rose 14 percent in September as a surge in federally assisted building and lower mortgage interest rates combined to put the long-depressed housing industry on an upward path.

The Commerce Department reported starts reached a seasonally adjusted annual rate of 1,146,000 units, the second highest level in any month this year.

President Reagan said the housing report "shows solid progress against the inflation and interest rates we inherited, sparking a rebound in this bedrock industry, a rebound that will benefit the entire economy."

In remarks before signing a proclamation making next week "National Housing Week," the president said the housing report was further evidence that his economic recovery program is beginning to work by reducing interest rates.

"We are beginning to succeed," he said. "The interest rates dropping sharply, housing starts rebounding, and our economic recovery program . . . I think is taking hold."

But while the information on housing starts shows the industry is beginning slowly to gain strength, a separate report on Americans' personal income, also released by Commerce yesterday, indicated the recession is continuing in other parts of the economy, especially in manufacturing.

Personal income from all sources rose in September by $7.7 billion, or 0.3 percent, to a seasonally adjusted annual rate of $2.6047 trillion, probably just about keeping pace with inflation, Commerce officials said.

Income from wages and salaries fell at a $400 million annual rate. In manufacturing alone, payrolls dropped at a $3.2 billion rate.

The drop in wages was more than offset by personal income from interest, which was up at a $2.3 billion rate, and from an increase in government transfer payments, mostly in the form of higher unemployment benefits, at a $2.8 billion rate.

The report also confirmed earlier retail sales figures showing that consumers stepped up their spending in September, primarily for durable goods such as automobiles sporting model-year-end discounts. Personal consumption expenditures climbed at a $20.2 billion annual rate, or 1 percent, to a seasonally adjusted level of $2.0076 trillion.

With their outlays rising faster than their incomes, consumers lowered their saving rate to 6.4 percent of their disposable personal income -- that is essentially income after taxes -- from 7.5 percent in July, when tax withholding was cut, and 7 percent in August.

Part of the improvement in housing was due to a sharp increase in starts being financed with federal assistance under programs Reagan wants to curtail or eliminate. Such starts jumped from less than 10,000 units in August to about 30,000 in September as builders moved equipment to sites and began work to make sure their projects would remain eligible for subsidies.

But the gains were more substantial than that. Single-family starts, which carry no subsidy, were up 6.7 percent to a 663,000 annual rate, the highest level in more than year. Housing experts said builders were stepping up construction activity in anticipation that mortgage interest rates, which have dropped two percentage points or so in the last two months, would continue to fall and boost housing sales. The number of building permits issued last month also rose nearly 17 percent, a sign the higher level of building likely will continue.

"Our feeling is that housing will continue to improve with interest rates declining," said Michael Sumichrast, chief economist with the National Association of Home Builders. Commerce Undersecretary Robert Dederick agreed, saying that as a result of falling rates "growth in home-building appears likely to assume its customary leading role in the economic recovery."

Mortgage rates are declining in the wake of a general drop in interest rates that began in July as the economy, which had appeared on the verge of a midsummer recovery, turned downward again. The Federal Reserve at first simply encouraged the market-dictated decline in rates, but when it ran its course last month and no recovery was evident, it at least temporarily changed its policy to push rates down some more.

Reagan said it was "no coincidence" that long-term mortgage rates peaked at 18.75 percent "the month before" his program started in October, 1981, and have now dropped to 13.38 percent. "Recent mortgage-rate declines have made home ownership possible again for over a million families," he added.

After he signed the housing proclamation, Reagan was asked how his comments squared with his veto of the subsidized housing bill Congress approved earlier this year. He replied, "We had the support of many in the housing industry in vetoing that because it was not going to be something of a real increase, the increase that we needed. And it was just going to further set us back with regard to federal spending without doing the job that we see now is being done by sticking to the course of our economic recovery program," he said.