Just exactly where do you intend to go, for how long, and which month(s)? 
Cheapest cities from the US are usually London, Paris, Frankfurt, but you’ll have to search for the best deals from Dallas for the specific time period (hopefully you’re at least a little flexible).  Sometimes there are good deals to Shannon or Dublin.  Icelandic air to Luxembourg is occasionally significantly cheaper than other options.  Usually it will be cheapest to fly to one of these cities and catch a flight on a European bargain airline from there to Spain, Italy or Greece.

Ireland, Spain, Italy and Greece is a pretty tough trip unless you’re going for at least 6 weeks — and then you’d just be hitting a couple of cities in each country.  You’d be better off to pick one of those 4 countries as the basis for your trip and then tour some other countries in that area.

Also, look at open jaw flights (maybe into Madrid, out of Athens) to save time. 

Well, that’s a natural question to ask, but nearly impossible one to answer. Price of airline tickets varies greatly from day to day. Even if Germany is, on average, cheaper then Spain, flying to Spain on a particular day may be cheaper then flying to Frankfurt or Munich on that same day.

I suggest you go to “Cheap Flights” forum and read the beginners guide (New to Cheap Flights? Start here.) Keep an eye on the flight deals that Don and others find and post over there. February is usually the slowest month for transatlantic flights, which usually results in some of the best fare sales. But you may have to wait until after Christmas to take advantage of those prices.

Considering that your itinerary is still flexible, and you are going during the coldest and darkest time of the year it may be feasible not to venture too far north. (That is to skip Ireland altogether unless you feel like you absolutely must go there.) Work your way from Spain to Greece (or other way round) using combination of trains and budget flights. Save the north (Ireland, UK, Netherlands, Belgium) for the next time.

I would start here: http://www.farecompa… … and look for places that are on your “must-see” list. Then I would go here: http://matrix.itasof… … to see if there is a good price available on a multi-city itinerary — to save backtrack time and expense. Once you’ve got your must-see list, post a question on the cheap flights forum next door and we’ll try to help you get sorted.