Goodeve was born in Middlebury, Connecticut. He moved to Los Angeles, California, in 1975. His earliest role was on a fifth-season episode of Emergency!. After a February 1977 screen test, he signed up as a cast member of Eight Is Enough, taking over a role played in the series' pilot episode by Mark Hamill.
When Eight Is Enough ended in 1981, Goodeve appeared in guest roles in series such as The Love Boat, T. J. Hooker, Dynasty and Fantasy Island, among others. In 1983, Grant did television pilot The Night Watchman. In the summer of 1984, Goodeve hosted Solid Gold Hits. In 1985-86, he played Michael James "Woody" Woodward on the soap opera One Life to Live. He also co-starred in the television movie Pigs vs. Freaks (a.k.a. Off Sides (Pigs vs. Freaks)) (1984). He also did several weeks a celebrity guest on the Pyramid game shows in the late 1970s and 1980s.
He reprised his Eight Is Enough role in two reunion movies during the late 1980s and also appeared on several episodes of Murder, She Wrote. As the 1990s came to a close, he also made an appearance on The WB's hit series 7th Heaven as Captain Jack Smith. In 2000, Goodeve appeared as the host of Word Pictures production Proving the Bible Through Archeology.
Goodeve moved to the Pacific Northwest in 1989 with his wife and three children. Soon after moving there, he began appearing in the recurring role of Rick Pedersen, an ill-fated bush pilot, on Northern Exposure. He also began contributing to KING-TV's Evening Magazine and starting hosting that station's travel show Northwest Backroads in 1998. As of 2014, Grant has hosted this show 16 years.
From 2000 to 2004 ,he hosted "If Walls Could Talk" and Homes of Our Heritage on Home & Garden Television He has worked as a voice actor for several video games, including the role of Wolf O'Donnell in Star Fox: Assault, the Engineer in Team Fortress 2, and various voices for F.E.A.R.. He also lent his singing voice to Bob Rivers's Twisted Christmas series of holiday CDs.
He is active in his Presbyterian church in Seattle, engaging part-time in an itinerant music ministry in the region.
In 2006, Goodeve appeared as George Bailey in Seattle's Taproot Theatre Company's production of "It's A Wonderful Life: A Live Radio Play". He returned to the Taproot stage in 2008's production of "The Christmas Foundling" as Old Jake.
He used to live in Washington state with his wife. He has a married daughter and a grandchild. He has temporarily moved to Georgia to pursue acting again.
Goodeve also appears in the Amtrak Cascades safety video.The Wonder of It All (1996)