Lot's of unknowing people answering your question. Over the years, we have spent at least six weeks in Green River. Now, our interests and activity level probably don't match the typical geezers our age.
We've always stayed at the KOA. Not a great place but nice. Free coffee and pastries every morning.The best part is the owner. He can give you lots of maps and advise on things to do.
The ATVing and Jeeping exploring the old uranium mines and scenic areas is absolutely wonderful. From just west of Green River you can drive SUV/Jeep/ATV south all the way to Goblin Valley. There are some routes through washes that are every bit as beautiful as nearby Moab. If you are a bit adventurous, you can even make you way all the way down to Capitol Reef NP.
West on I70 and north of Green River are some "roads" that will take you to one of the most phenomenal Native American pictograph images in Utah. There is another drive you can do in any vehicle to an old suspension bridge and north and back east to the Provo highway.
The museum in GR river is wonderful. It documents the exploration of the Colorado by Powell.
There is some of the best hiking in Utah just west of GR in the San Raphael Swell. The Black Dragon is very memorable but a bit of a grunt. Lots of other short hikes on the way to Goblin's Valley.
The town of GR sucks. There is a small very expensive grocery but a good Ace Hardware. No food worth eating except maybe the Chesters Chicken at the bus stop.
If you are just car bound tourists, two to four days should do. Remember to ask the KOA owner for suggestions.