When hiking in the backcountry always let someone know where you are going and when you expect to be back. Avoid hiking rainy days if hiking Ashdown Gorge or any other slot canyon.
fees/permits
If visiting the Southwest where there are so many parks and monuments so close to each other, get the National Park pass. This pass is good for every National Park in the USA and it gets you into Cedar Breaks National Monument as well. If you want to pay just the fee for the monument then it is $4 for 7 days.