Do Barr Trail if you can only do one.
Tips:
1) Get going early, like 7:00am! You want to be done very early in the afternoon, you don't want to be caught in a thunderstorm above A-Frame...
2) Pack plenty of water/snacks/gloves/hat/jacket. There is water available out of No Name Creek and the stream at Barr Camp but you'll probably want to filter it. (I take may own water on training runs so it makes race day even that more special by not having to lug all that extra weight up)
3) Plan your way down.
- You can get tix for a ride down on the Cog but if you miss your scheduled time you're SOL, see next points...
- Hitch hike, usually not too hard to get a ride back down to Manitou from the Peak.
- Have somebody pick you up at the top.
- Walk/run down the trail (points 1 and 2 above apply), but your quads and flexors will probably feel like crap for the following week or two.
There is also a pretty good possibility of significant snow on the trail above A-Frame mid June, be prepared.
Here is a link to a camera at the top, check out the piles of snow either side...
http://www.cograilway.com/Summit/summitcam.jpgSnowed today and will probably snow up there everyday this week!!!
2 years ago there was ~6 feet of snow on a small section of the trail near the top 2 weeks from race day. Thankfully, volunteers ice picked and shoveled it away.
Good Luck!