Check out bushducks.com for trail conditions. Click the "high pass status" tab, and scroll down. All of the Colorado trails are there. As stated above, many of the higher ones will still be blocked by snow.
As for which trails to try and hit- the San Juans are the icon in Colorado.
I would say Red Cone would be high on the list, but it tends to open late.
Holy Cross- and as an alternative to Holy Cross, check out the McCallister Gulch/Wearyman Gulch loop. It was one of my favorites in 2016, and has awesome views of Holy Cross, without the difficulty.
Schoefield Pass/Devils Punchbowl/Lead King Basin might be open. The wildflowers on Leadking in late June are epic. Probably my wife and I's favorite trail to date.
Hop on amazon and get the "guide to Colorado trails" by Charles Wells. Best book out there IMO. They have a Northern and Southern version. Not much in the Northern version though.
Good luck!