http://www.aranislandferries.com/ runs from Ros a'Mhíl west of Galway to Aran and to Inis Oírr and Inis Meáin. There are busses from Galway to Ros a' Mhíl. The company itself also runs a shuttle to Ros a' Mhíl from Galway.
It is not so easy to get around Conamara by public transport. However, you can check bus schedules on http://www.buseireann.ie/inner.php?id=238 - scroll down for counties Galway and Clare. You can also view the whole network map here: http://www.buseireann.ie/pdf/1282732129-Network-map.pdf (big and slow to load).
If you want to see Conamara, route 416 will take you to Carna along the coast. Route 424 will take you the same way but then branches off down to Leitir Mór and Leitir Mealláin. Route 421 will take you via the inland route to Roundstone, Leenane and Westport.
Route 423 will take you down to Doolin and the Cliffs of Moher.
Be careful reading the timetables, as not all the services run all the time. For example, there is a bus marked as coming back from the Cliffs of Moher to Galway at 1850, but this is only from 27 June til 25 September, so if you are going next weekend it will be no good to you.
I think you have basically three destinations - Doolin and the Cliffs of Moher, Aran, and Conamara (either the coast or via the inland road to Clifden). In two days, you'll only have time to see three of them. There is a ferry from Doolin to Aran, so theoretically you could combine Doolin, the cliffs and Aran, but you won't have enough time to do that in one day if you are relying on the last bus back at 1430.
However, given how little time you have, it might be worth seeing if there are any organized tours run from Galway that would allow you to see some of these places more efficiently.
