My daughter and I are flying into Denver on May 11th and leaving May 17th. We are avid hikers, love nature and adventures, and do not like anything too touristy. We've been to the majority of the National Parks, but not RMNP. I need to book lodging and plan hikes, but with this time of year, snow, seasonal road closures, and altitude acclimation, I feel stuck. I originally thought we'd split our time between RMNP and Indian Peak Wilderness, but maybe it's not the best time of year for that. I also thought maybe we'd be better off heading west towards Glenwood Springs to hike Hanging Lake (I can still get a permit), rent bikes one day, and then do hikes and visit towns along the way back to Denver. I thought about driving down to Garden of the Gods, but we've been to Zion (hiked Angel's Landing) and Bryce so that may not be as exciting for us.

So, as you can see, we are open to heading in any direction in our rental SUV. We live in Charlotte, NC, so will need to adjust to elevation. Also, I am ready to buy spikes for our boots...we can rent snowshoes - our goal is to see as much natural beauty as possible and challenge ourselves along the way. I would appreciate any suggestions!

Mid-May can see snowstorms in Colorado, so make a plan with indoor contingencies in case of inclement weather.
For a mother-daughter getaway, would you like a gorgeous mountain ranch resort, with all the outdoor activities you want, plus a spa, pool, amazing service and food? A very nice place you could stay for the entire trip, to relax and rejuvenate? Horseback riding, hiking galore, zip lining, fishing, day-trips, special events hosted for guests at the Ranch. In case of bad weather, you can have a spa day or sit by the fireplace in your room, and fireplaces inside and outside the resort buildings. You would be able to do things separately, if you want.
Our favorite is Devils Thumb Ranch, in Grand County.
https://www.devilsthumbranch.com
DTR is pricey, but worth it. The owners have created an eco-friendly ranch, with homegrown cattle, vegetables, horses, etc. Interiors are authentic Ralph Lauren-esque, with actual Western artifacts, but not corny or overdone. Customer service is outstanding. I would not consider DTR 'touristy', given its wide open spaces and somewhat remote location. It is a stunning property, which goes as far as the eye can see.
DTR is offering special springtime rates: "Receive your third night free when your stay falls between Sunday and Thursday nights, or your fourth night free when your stay includes a Friday or Saturday night. Based on availability from April 6 through May 27, 2021."
P.S. I doubt if you need spikes on your shoes, if you have good treads on them. You can rent whatever gear you need at DTR.

If rental cars are unavailable or very expensive in Denver, you can always take the Amtrak California Zephyr or the Bustang to Glenwood Springs. There is a great hiking/biking trail east of Glenwood along the Colorado River. There is also a good bus service up to Aspen with a side trip to Maroon Bells.

As MoBro said, highly unlikely that you'd need spikes this time of year. The snow will mostly be too soft, exception of north facing slopes early in the morning. Snowshoes could actually be handy if you were at higher elevations of Indian Peaks or RMNP high country. That said, popular trails are usually packed down pretty well.

Lower elevations, say ~8,000', might be muddy or dry, hard to say, but at least have some good gaiters. Hiking in the foothills elevations, 5,000-7,5000 feet is usually excellent now.

Tom's idea of Glenwood is not bad.