I used the Familysearch site to find the marriage:
Then there’s a way of finding out who the partner was using the Ancestry site:
You enter the year, location and then page number (taken from the reference found on the familysearch site) in the keyword box, and it brings up a list of possible spouses who married in that place at the same time. In your case there were two named Rose so I can’t tell which is the one your ancestor married. (Though you’ll find that out if you order the certificate).
Statutory birth registration started in Ireland in 1864 so you won’t find a birth cert for your ancestors from Clare. You might find their baptisms but to do that you really need to know their religious denomination and parish. Statutory registration of Non RC marriages started in April 1845, but RC marriages were only registered from 1864, so if the family were RC, the only record of that marriage will be in parish records, if they survive.
I don’t think any parish records for Clare are on-line yet. I’d assume Michael Mooney was RC. If so, you can see what parish records survive, and where they are held, using this site:
Unfortunately there’s about 30 parishes in Clare, and unless you can narrow the search to the right one, it’s a needle in a haystack.