Good morning Luis,
A very Happy New Year to you, welcome to the forum.
So you have a fairly tight timeline to fit in everything you want to do, especially on public transport, but it might just be do-able! I will the leave the Connemara part to one of our other community experts, and stick with the part I know best - the Cliffs of Moher :)
In fact, not only is it very possible to take a tour
to the Cliffs of Moher, (and get to see them from the sea on a boat trip and at the Visitor Centre) you can also get to learn about the amazing Burren area in North Clare AND take a half-day trip to the Aran Islands, before taking the bus back to Galway. I would suggest you do this on your second day, as it is an early start from Galway and gets back to Galwy for approx 19:00. There is a late bus back to Dublin from Galway (20:15) with Citylink
Here's a little video of this tour to give you a better understanding of just how wonderful our Wild Atlantic Day Tour
You can read more about our young intern Niamh's adventures with her friends in her blog here
and reserve your spot
here if you wish.
I hope this helps with your planning, Luis, let me know if you have any questions, and enjoy planning your adventure!
All the best,