Veteran Jamaican defender Goodison has signed for a further season and the deal, which is subject to work permit renewal, will take the 39-year-old into his ninth year with Rovers on the back of 378 appearances.
Wales Under-21 international Taylor has played 110 times since first agreeing professional terms with Tranmere in December 2008 and has been rewarded with a fresh two-year contract.
Manager Ronnie Moore said: "I'm pleased both players will be with us again next season.
"Ian has been a rock for Tranmere over the past eight years. He plays through pain, leads by example and has an incredible will to win. He is still one of the best defenders in the division and centre-forwards know they've been in a game.
"I've been really impressed with Ash since I came back to the club. He's a strong defender who has a bright future ahead of him. They've formed a good partnership together and hopefully it will continue next season."