Some homeowners aren't breaking out the champagne after they've closed on their new home purchase.

Nearly half of the homeowners in a recent NerdWallet survey said that they would take a different approach toward buying if they were going through the process again.

The personal finance website polled 2,241 adults in January.

"One thing I'd advise — and no buyer really follows it — is to shop around and do more research not just for your loan but for the home," said Tim Manni, who covers mortgages at NerdWallet.

Younger homeowners — millennials and Gen Xers — expressed the most buyer's remorse after closing on a new dwelling.

About 3 in 5 of these participants said they had regrets throughout the shopping and mortgage process. What they lament the most is that they should have saved more money before buying and that they should have shopped around for a mortgage.

The difficulty of saving

A fifth of all the participants said they wished they had more in savings before buying. Nearly 3 in 10 millennials echoed that sentiment.

As housing prices continue to rise, prospective buyers have a harder time finding the necessary cash for a 20 percent down payment — the amount you'll need to avoid the additional cost of private mortgage insurance.

The median listing price of a home in the U.S. is now $234,900, up 4.4 percent year over year, according to Zillow. Now that the spring homebuying season has kicked off, high demand and skyrocketing prices are leading to aggressive bidding among prospective buyers.

It doesn't help that some markets are extremely competitive. A recent study from Bankrate ranked California as the toughest state for first-time homebuyers, citing limited inventory and lack of affordability.

The median price of a home listed in the Golden State is now $460,844, according to Zillow.

Banks, including Wells Fargo, have launched mortgage programs that will make loans available to first-time buyers with as little as 3 percent down — provided these borrowers have strong income and attractive credit profiles.

Federal Housing Administration loans also allow buyers to borrow with as little as 3.5 percent down, but those borrowers will still have to pay mortgage insurance. President Donald Trump recently blocked a mortgage insurance premium cut for FHA borrowers, further raising costs for those buyers.

Don't forget that affordability is largely a function of your cash flow each month: What can you devote to paying down your mortgage?

"Three percent down will get you in the door, but it will make each month a little harder," Manni said.

Housing costs, including taxes, principal, interest and insurance, should not account for more than 28 percent of your gross monthly income. This is known as the front-end ratio.

At the same time, the monthly payment for all of your debts generally should not exceed 36 percent of your gross monthly income. That's the "back-end ratio."

Shop around for your mortgage

More than 4 in 10 of the polled participants who applied for a mortgage felt they weren't aware of all their loan options.

It may seem convenient to go with the loan professional your real estate agent suggests, but Manni suggests that you look at a range of lenders and see what they offer.

"It's your responsibility to ask 'What's my timeline? Do I need a 30-year mortgage? Do I want to avoid adjustable rate mortgages altogether?'" Manni said.

You should also understand the details of the loan recommendations your lender will put together.

For instance, some banks offer a "no PMI" mortgage, which allows you to put down less than 20 percent of the purchase price without the additional cost of mortgage insurance, Manni said. But you're paying for that convenience in the form of a higher interest rate.

Something else worth asking as you do your research: Are there mortgage points? Points can lower your interest rate, which will cut your monthly payments. And, you may be able to deduct them on your taxes.

"We'll see rates go up, so it might be worth paying the points," Manni said. "That's the crux of the issue: People wish to save more. Do you put it toward your down payment or toward your points?"