By airport, I am assuming DIA (Denver International)-- do you happen to know the town?
I am just asking because the airport is a good 30-45 minutes from downtown Denver and then the Foothills are another 20 minutes or so depending on where you want to go. So if you really are right by the airport, you are going to have a good hour commute just to get to the foothills.
Another question is what is your ability and do you have snow gear (ice axe, crampons/microspikes etc)? Because even in May, most of the high country is still going to covered in snow and so I generally bring my basic snow gear with me at least through the end of June and into July.
Some good solid choices between Denver and the Eisenhower Tunnel (I-70 corridor):
-Beaver Brook Trail (9miles one way/elevation gain is perhaps 1,400ft)
-Bergen Peak, Evergreen. 10miles RT, 2,700ft gain.
-Squaw Mountain and Chief Mountain. These two are very short hikes, perhaps 5 miles and 1,000ft. But great views and very close to the city.
-Chicago Lakes. This is one long day, but worth it if you like high-country lakes. The downside is that May 1st could mean that it is still frozen and ice-off hasn't yet begun.
Personally, if you feel up to it, then I would drive a little further west to Loveland Pass and hit Mt. Snikatau. It is probably the most accessible 13er (paved road all the way to the top of the pass) and is ranked one of the easiest to complete. So if you want to get up above tree-line and do some high altitude hiking over easy class 1 terrain then this is the summit to do and the views are incredible for being so close to the I-70/US-6.
Heck, when the time gets closer and if you decide that you want someone to tag a long, then shoot me a PM and depending on my schedule I wouldn't have a problem taking you up some of the easier peaks, that is if summits are what you are interested in.