Homeownership wasn’t exactly on the Tinklenberg family’s radar last year. Evan Tinklenberg had just finished graduate school in Chicago and landed a job as a campus pastor at Geneva Campus Church in Madison, Wis. His wife Kim, a nurse, was starting a new job at the University of Wisconsin’s cancer center.

They had always rented and figured they had years to go before they could afford to buy anything, especially in Chicago. Madison would offer more for the money, but with two children under 5 and a big dog in tow, they found that three-bedroom homes there were asking well over $3,000 a month.

So the couple, now both 30, decided that if they were going to live in this home for a while, they might as well put their savings to use. “Our life has been so transitional for a number of years,” Mr. Tinklenberg said. “This is the first time we’ve actually had a chance to do that.”

They moved into a temporary furnished apartment they found through Furnished Finder and got acquainted with their new city. Home prices were in their range, which topped out at $350,000, but as in much of the country, inventory was slim.

Their broker, Brittany Trent of Keller Williams Realty, assured them that prices were starting to thaw. “Kim and Evan bought at a pretty good time,” she said, adding that “patience and an open mind” always help in a competitive market.

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The couple’s non-negotiables included living on a quiet street, and no subdivisions. “We play outside a lot, so we wanted more of a neighborhood feel,” said Ms. Tinklenberg. And they wanted an easy commute downtown, especially after driving an hour or more to get to work from their temporary apartment.

“We wanted something with more character and families that had been here for a while,” Ms. Tinklenberg said.

They looked at dozens of homes online, then narrowed it to a handful to tour in person.

Among their options:

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