The Galway Advertiser is one place you can look for accommodation, and is best to view the classifieds section online as it gets updated just after midnight Wednesdays. The Galway Advertiser is a free paper available on Thursdays but often come back to college time, there is the accommodation list available from their office on Eyre Square for a fee and after a long queue. Their map isn't to scale so you can always get a map from the tourist office or through Discover Ireland (maybe anyway)
http://www.advertiser.ie/galway/classifieds/
You're bound to find something on the likes of Daft.ie and Let.ie but I would recommend checking some of the other local papers (Galway Independant - another free paper, the Sentinel (around 50c or so) The Galway city tribune think around €1 or 2 but check if have classifieds first...galwaynews.ie), as well as local supermarkets and shops, or even put up a card in a shop yourself. You could also check some of the Student Union notice boards in GMIT as well. I think the accommodation office still does those small packs that give the listing of digs, but you can probably find that on their website.
I assume you mean GMIT on Dublin road and not the Chluain Mhuire campus btw
I live near GMIT as I studied there some years ago, so you've plenty of choices for areas to live in such as Murrough, Merlin Park, Renmore (which has the old student rentals of Lurgan Park, Glenrua), Ballybane, Castle Park, Ballybrit, Mervue that are all within walking distance of GMIT. Wellpark is a little bit further down the road by the Eye Cinema, but it's still reasonably close to GMIT.
Even though student villages are not an option for you, beware that one student village, Glasan is right across from GMIT gates and the other Gleann Na Ri is behind an estate called Lios an Uisce, behind Kingsvalley Hotel at Galway Crystal.
Some of the older houses in Renmore and other places can be a bit of a dive, so make sure you know what you're entitled to get for the money so you don't get ripped off, as well as know your rights. I've met some seriously dodgy landlords in these areas and in the city. I've found that from house hunting in the last year, some landlords like to tell tall stories that their previous tenants upped and left, did their own diy of changing the interior and electrics and put a lot of pressure on me to take the places I viewed. Make sure you know what the going rate is and haggle.
I would highly recommend that you take a walk or cycle around the areas first because there's a great variety and all the areas I've mentioned are just off Dublin Road and not far GMIT (with exception of Ballybane and Ballybrit but are still near. There may be pitfalls to a lot of places, so ask for as much local information as you can. Renmore
and some parts of Mervue (from Dublin road side) are quite green and leafy with nice amenities near by. a friend of mine (also a mature student) moved to Mervue some months ago and when visiting him I found a really lovely community, lovely gardens, nice people and an actual sense of community.
I'd also recommend posting the same question in the Galway Forum on www.boards.ie or do a search of the forum for some ideas on areas and some feedback, as I recall there have been posting about this before. I recall there was a thread about rental price on that forum a short while ago, but the last link below is a search of 'rent' and 'rental prices', just to give a start.
EDIT:
just thought I'd add, if you're in Galway city you can take the No. 9 Parkmore bus at the bottom of Eyre Square (Opposite O'connell's pub) that will take you to GMIT side entrance on Dublin Road. They go every 15 mins until 7 or 7.30, but you can look up the galway city timetables here http://www.buseireann.ie/inner.php?id=59#galway