Connemara is geared for tourists, so finding accommodation is not difficult. Frankly, I'm not able to advise you when it comes to hotels - they tend to stay on the expensive side and are far outside my budget (if you manage to find a room for less than 50 euros a night you can consider yourself lucky).
Camping is another matter. I've camped all over Connemara, both on campsites and rough - it works.
Campsites and caravan parks are very easy to find. Ask anyone local or look for signs, there's bound to be a camping field in each larger village. It costs around 5 euros a person and this price usually includes access to some sort of bathroom facilities.
Camping rough is officially frowned upon (especially by campsite owners). Not that I recommend breaking rules, but I've camped rough many times and never had any problems. I generally follow the guideline 'be no trouble and no one will bother you' - you know, clean up your rubbish, don't make too much noise, stay out of sight and nobody gets upset. Some people can get quite angry if you trespass on their land so try not to jump too many fences, or, if in doubt, ask if it's ok to camp.
Generally, camping in designated spots is risk free, but when it comes to ambience, nothing beats pitching a tent on some desolate beach without a soul around.