(Note: this travelogue assumes you are starting and returning to Galway in the same day.)
First, hire a car and start early, bringing raingear, walking shoes, and a picnic. Then, for the first stage, take the Clifden road out of Galway as far as Maam Cross (passing through picturesque Oughterard and crossing the Upper Corrib on the way).
Most guidebooks recommend that you continue straight on to Cliften from here, but I think you should turn right, heading instead for the tiny hamlet of Maam. There's nicer scenery this way: sullen hills and recumbant bogs. But please take care not to get distracted by the views: the road is windy and narrow!
At Maam, you can break your journey and dawdle at the pub over morning tea. However, I'd recommend turning left and heading for Leenane, through the long valley that runs between the Maamturk Mountains to your left and the smaller foothills of Joyce Country to your right. Leenane is where 'The Field' was filmed. It's a small village, more noted for its fine, scenic location than its amenities, though there are a couple of pubs there where you get brunch.
From Leenane, continuing south on the Clifden road, you will pass by Killary Harbour to your right; stop and admire the view before moving on. It's spectacular. There are many other potential places to stop on the Leenane-to-Clifden road: Kylemore Abbey, Letterfrack, the Connemara National Park being just the most famous. I'd recommend you see them all but, if you are stuck for time, Kylemore is your best bet. As for the rest, just enjoy the drive to Clifden, and make a note to come back some other day.
In Clifden - a large town overlooking the moody Atlantic - you have a choice to mull over while you lunch: whether to go straight back to Galway through Maam Cross and Oughterard, which will take you about an hour-and-a-half, or you can continue south again, along the convoluted road to Roundstone. If you started early, and have the time, I'd take this road. It will lead you through some of Connemara's wildest and prettiest scenery, and eventually to Roundstone, a nice, though admittedly tourist-oriented, village.
For your final stage (it's probably early evening by this stage and you're tired) head for Casla, then take the Spiddal road, along Connemara's southern coast. To your right there will be a view of north Clare, beckoning you towards the Burren, if you haven't already been (and maybe even if you have). You'll be back in Galway in time for a bite to eat and a pint before bed.
(PS. Of course, to do all this properly, it would be best to take a couple of days and break your journey down into three or more stages, staying in Leenane on the first night, Clifden on the second, and Roundstone on the third. That way you'll not only see Connemara, but get to meet it's people, and maybe even hear a word or two of Irish spoken. Remember, 'dia dhuit' (pronoucned de-a which) is hello, while 'slán' (rhymes with lawn) is goodbye).