Is Ireland good without a car?

The places you mention in Ireland are all in the west, where there is really no train transportation. That only leaves you with the bus. You could base in a couple of cities and take day tours from there, but you'd really be much more flexible with a car. Since you have plenty of time, you might start watching how the prices run so that you'll recognize a good deal when you see one. If conditions permit, you can book whatever looks good, and change your reservation if you find something better in the meantime. Many companies let you cancel or change a reservation without a fee.

As for your plan to fly into Dublin, since the things you want to see are all in the west, you might consider flying into Shannon instead. If you do decide to rent a car, it's a lot easier there to get used to driving on the other side since Shannon airport is out in the middle of nowhere instead of on a major street like the Dublin airport.

It depends on how badly you want to go to the Southwest Coast. A car is almost essential for Dingle and Ring of Kerry. If you focus on Dublin and Galway and take a few day tours then not having a car won't affect you much.

Having said that, I'd recommend the car. The Southwest is one of the best part of Ireland to visit. For a week, you should be able to get a car with automatic transmission for about $700-$800. I know you're traveling cheap but two months in Europe must be costing a heck of a lot more than that. It's worth spending a little more to get the most out of it.

I echo the posters who say a car will be easiest but it is possible to travel on public transport with a few caveats.

The Irish train network is not the most comprehensive, but don't necessarily dismiss it out of hand. It doesn't matter where it doesn't go as long as it goes where you want to. Timetables can be had from the natiional rail company http://www.irishrail.ie/

If the trains are out that leaves buses. This is easier in Ireland than most countries as the bus system is nationalised and one company runs local and long distance buses (with the exception of within Dublin.) This means you can get timetable information across the country easily from http://www.buseireann.ie/ they also have a journey planner.

The one ting I'd point out is that travel in rural areas can be quite slow by public transport, not necessarily because of journey times but service frequency. Remember to factor in waiting for the next bus when you calculate travel time. Be careful that you don't save money on a car to allow you stay longer and then waste all that extra time waiting at bus stops.

We rented a car in Ireland this summer, it seated 5 people (and we did fit 5 people in there, including car seats), and cost a little less than 200 euro for a week, including insurance. It was a manual, not an automatic. We got ours through e-car rentals Ireland, they're a consolidator, the car was actually with Thrifty. Remember to factor in gas. We had a diesel, and only had to fill it up once, driving from Dublin out west and back to Dublin.

Well, actually, why don't you take a train from Dublin to Galway, and then limit your driving to be from Galway to Dingle (personally, I'd leave off the Ring of Kerry). Drive back from Dingle to Shannon airport for your flight. This may help stretch your budget.

Scotland and Ireland are amazing. Do not miss pub music and the west of Ireland. Dublin is nice, but as Rick says, go west!

I spent seven weeks in Ireland this past summer and only used train/bus transportation (including the southwest area). I did wish I had a car at times, but overall I did just fine with public transportation and then taking day tours places. I liked day tours because you learn a lot on them with the guides and meet other travellers that you wouldn't do driving yourselves. The buses do only run at certain times/days so you definitely need to check a few days ahead to make sure you can actually get to the places you want to go.

That being said, I did find the buses quite a bit more expensive than they are in Canada. It ended up costing me about 30-50 dollars to take the bus for just a couple of hours which surprised me. There is also the price of day tours then as well (usually between 50-60 dollars). If there are two of you, then it may end up being more cost-effecitve to drive - I'm not sure. You can check out the rates on the bus site that was already posted.

you would be very limited w/o a car. Driving/parking is not complex in Ireland except for big cities.

Just returned from 11 days in Ireland with my preteen daughter and am pleased to report that you can indeed have a "full on good Irish time" using only public transpo, it was a breeze, but essential to research and of course do not pack more than you can carry, it makes ALL the difference in ease and enjoyment.
Bus Eireann is heavily used by locals and tourists because it has a very good system that covers all of Ireland. Their Network Map and schedules are available on their website in PDF. Study them and plan accordingly. They run constantly, so you can always change plans. Passes are customized to your needs; we purchased 2 passes (1 adult 1 child) good for 7 days travel over an 11 day period for less than 200 euro (purchased in Limerick). Buses are clean, comfy and easy to use.
We also took 2 trains(Cobh to Dublin, on our way to Newgrange, and Dublin to Ennis, on our way to Doolin; total cost:60 euro) Irish Rail also has maps and schedules on line. They offer cheap fares when you purchase mid-week, keep checking prices and buy on-line when low (also check different departure times for cheaper fares) Trains were also clean, comfy and easy to use.
By the way, we found Dingle very do-able without a car. Like most towns, the bus stops there several times every day, its small enough to walk everywhere in town and you can walk, bike or hire a driver to do the essential Slea Head tour.
You seem to share our interest in history and the outdoors so I highly recommend trips to historic Newgrange (90 min N of Dublin) and the picturesque village of Doolin (settle in, then take the 20 min bus ride to the Cliffs of Moher, and walk 2hrs downhill back to Doolin for pub-grub, Guinness and trad at O'Connor's Pub)
As for traveling cheap,check out hostels, most offer private rooms and many provide a substantial "continental" breakfast to save even more.
Public transport will get you anywhere you want to go cheaply and hassle free.
You'll L-O-V-E Ireland, Cheers!