Hi Vicente, welcome to the forum!
Sounds like you only are here for a week, with a car, is that right?
Since I'm not much of a mountain biker or hiker, I can't provide very specific suggestions, but I'll chime in what I do know.
Rocky Mountain National Park has fantastic hiking, but you are not allowed to mountain bike off road in National Parks, and that includes Yellowstone and The Tetons. The parks are all very worth visiting, especially for hiking, but you can only bike on the roads and bike paths. Late May is really early season for both RMNP and Yellowstone. Trail Ridge Road (across RMNP) opens around Memorial Day (5/25-ish) and closes mid/late October for the winter. Yellowstone services aren't all open yet in May either.
www.nps.gov - national park service web site.
So, I guess I would lean towards staying in Colorado.
Summit County (cities of Frisco, Dillon, Breckenridge) has a lot of both biking and hiking/climbing, but is heavily populated. Leadville, about 30 miles south, is a hub of outdoor mountain activity, and less commercial. If you end up there, stay in the hostel (
www.leadvillehostel.com) they are super cool and you'll meet people doing the stuff you want to do. One problem for you with Leadville is that it's very high, so if you have trouble adjusting to the altitude, you might want to change plans and go somewhere lower (everywhere else is lower).
The high mountain trails (Leadville) will probably still have some snow in May. I would look at going to Durango, it's a mountain bike destination and surrounded by gorgeous mountains, less populated, and I'm sure there are tons of places to hike. Crested Butte is another Colorado MTB destination, but I have a feeling May is too early, June may be too early even.
Fruita, up near Grand Junction, is another MTB destination. It's warm and pretty low there - May should be nice and warm, maybe slightly too warm.
It will be too hot to mountain bike in Moab in June - if you want to go there -- it is spectacular -- go there as early as possible. I rode the White Rim Trail near Moab 2ce - early May was nice, and late May was too hot.
Late Sept/Oct would be good for mountain biking in Moab and Fruita, not sure when the Durango season ends, but probably by mid October it's too late. The hiking in the high mountains will be very cold, and expect snow in late Sept/October. The main road through Rocky Mountain National Park usually closes for the season mid/late October.
In terms of getting more info, all of the towns I mention have one or more bike/hike/mountain shop, and I would ask for local information when you're there, that's the best way to figure out exactly what trails are in good condition right at that time. I'm sorry I don't have anything specific, hopefully you'll get a bunch of other responses.
I guess if you only have a week, I would focus on finding out more information about Durango and Leadville - and if you go to Durango, Moab is really close.