Dublin to Shannon?? 5 nights in Ireland HELP

We're planning on visiting Ireland for our next trip, and then will be flying over to Scotland. TripMasters.com has incredible fly/drive packages for Ireland--see their website for itineraries. The prices right now are very reasonable.
We might stay in Dublin for a weekend, but will then hit the road to stay in small hotels and/or bed & breakfasts in small towns. And we'll be hitting the local pubs there.

Is this your first trip to Ireland? If so, and if you're at all averse to driving, then you could easily spend all of your time in Dublin and not run out of things to do. The city itself is worth a couple of days of exploration, and there are numerous day trip options to get you out to places like Powerscourt, Glendalough, the Boyne Valley and Newgrange, the Hill of Tara, etc., and a day trip up to Belfast is only a couple of hours away by train.
That said, exploring the west of Ireland by car is always a great way to go. You could conceivably spend a day or two in Dublin, then take the train (3 hours) out to Galway, pick up a car there and enjoy Connemara for a couple of days before driving down to Dingle or Killarney (by way of the cliffs) before dropping the car at Shannon for your trip home. With only 5 nights to work with you'd really need to prioritize to keep from running yourself ragged trying to check things off a list. It's a matter of personal preference but I'd suggest spending at least a couple of nights in a place and using it as a base for exploring the surrounding area. Both Galway and Dingle/Killarney would be great for that.

I agree with everything Robert said. With only five days you will have to prioritize, and possibly take some things off your list. You could easily spend all of your time in Dublin and not have to worry about a car rental, learning how to drive on the left (which is easy, but a little stressful at first), and then finding your way. You can easily take the train to either Killarney or Galway, and both towns have many options for guided day tours you can do. Personally, with so little time I would skip Cork (Blarney Castle is just a tourist trap in my opinion), go straight to Killarney, and from there you can get tours to Ring of Kerry, Dingle, Gap of Dunloe and many other places. If you take the train to Galway you can easily get day tours to the Cliffs of Moher/Burren, Connemara region, and Aran Islands, but from Galway you will either have to return to Dublin, or go down to Shannon to fly home. Without a car the problem you will have with so few days is the time you will use getting to either Killarney or Galway in time to join a guided day trip. Most of these all day trips will leave at 0900, or thereabout, and if you aren't already in town you will miss it. If you can't add a couple days then the simplest solution might be to stay in Dublin as that would allow you to maximise your time. If you are set on seeing one other town then I wold pick Galway as it is closer by train.

So we will probably go to Galway since it is closer (distance wise). Would you recommend train or renting a car? If we rent a car, will be miss out on the "day tours" since we will be driving to/through all these places on our own?

If your preference is to take day tours once you're in Galway then you really don't need a car - just take the train over, settle in, and book whatever tours appeal to you.
We've always appreciated the freedom and flexibility of having a car in Connemara. There's a scenic drive by way of Clifden, the National Park, Kylemore Abbey and the village of Cong that's quite pretty, and a day out on the Aran Islands would be fun too. For info, the Aran ferry company has a pickup point right in Galway that will take you out to the terminal at Rossaveel for the trip over to Inishmore so, again, you could easily get by without a car if you so choose. It really comes down to your personal preferences.
Having a car would also make it pretty easy to get from Galway down to Shannon airport - about an hour and a half as I recall. Am sure there's a bus that would get you there too if you opted to go without a car.

So if we take the train to Galway, we can still rent a car in Galway? How close is the rental car place? Is there a fee to return the car to Shannon? I can't find this option on Budget rental car!!!

Yes, you can rent a car in Galway if you take the train over from Dublin.
You may need to widen your search for a rental company - Budget's rates look pretty punishing and I see that they do hit you with a one-way drop charge. Based upon a quick look at Travelocity it looks like Payless is the cheapest way to go if you rent in Galway and drop off at Shannon, but you should investigate that in greater detail keeping in mind that the matter of the insurance will be another consideration - most US credit cards don't cover you in Ireland, and those that claim they do have gotten mixed reviews from others on the forum.
Not to further complicate your decision making, but the default rental in Ireland is a stick shift, and a left-hand shifter at that. You might want to consider specifying an automatic if it will be your first time driving on the left. There's usually a larger selection of vehicles to be had (and thus a better chance of getting an automatic) if you rent in Dublin. With a more limited selection of vehicles available in the countryside it could be more difficult to guarantee an automatic.
Dan Dooley http://www.dan-dooley.ie/car-rental-hire-galway.htm has consistently gotten good reviews from others on the RS forum - the rates are fair and everything (including insurance) is clearly explained up front. Plus the pick-up location is on the outskirts of Dublin so you wouldn't have to drive in the city... never a good idea IMO. I see that he has a pick up location in Galway too. Your other option would be to pick up the car from a Dublin airport location. The drive over to Galway is motorway all the way.

Thank you! I have reached out to Dan Dooley for price quotes from both locations, but it looks like leaving from Dublin will save us some money! We are planning on driving from Dublin to Galway. Do you recommend any pit stops along the way?

The last question is do we stay in Galway for 3 nights and make days trips to places, or do we travel south and spend a night in Dingle and then go to Killarney? Again, if this seems like too much in a short amount of time let me know! Our biggest site we want to see if Cliffs of Mohr but we are open to others!

With only 3 nights at your disposal I probably wouldn't recommend trying to include Killarney or Dingle into your itinerary, else you'll wind up spending most of your time on the road getting from place to place. The drive from Dublin to Galway takes about 2.5 hours via the motorway, so with a reasonably early start you could incorporate any of a number of stops along the way and still be in Galway in time for a pint. In no particular order (and without knowing your specific interests) here's a short list of possibilities:
About 45 min. north of the airport are Newgrange, the Boyne Valley, and the Hill of Tara - lots of Irish history condensed into a small area. Wouldn't be too far out of your way.
About an hour south of Dublin is the Glendalough monastic site. It's situated in a beautiful forest setting, with a number of easy to moderate hikes if you're looking to stretch your legs. Recommend getting there early - ie before the tour buses start to roll in at 9:30 - to have the place all to yourself and be able to wander around the ruins at your leisure. Need to be aware that the bathrooms are located inside the visitor center ... which doesn't open until 9:30 either.
Another monastic site closer to your route is Clonmacnoise. Fewer crowds than Glendalough.
A bit out of your way, but maybe worth it, is the Rock of Cashel. Always worth a stop IMO, as would be Kilkenny and its castle.
Could also just detour a bit south on your journey and incorporate your visit to the Cliffs that day, and then meander up to Galway once you're done.

Depending upon what time your return flight leaves from Shannon you could either spend your last night in Galway and drive to the airport in the morning, or drive down the evening before, drop the car, and find somewhere close to the airport to spend your last night. Haven't personally stayed there, but others have recommended the nearby Bunratty Castle and Folk Park - both of which appear to be great fun. Could then sleep in the next morning before heading to the airport.
No matter how you organize this it will still be great - its all Ireland. I would just caution against trying to cram too much into your itinerary. If there's an experience that's meant to be savored it's Ireland. Assume that you'll be returning one day to see things you missed on your first try - the place does have that effect on people.