Edinburgh, Scotland to Shannon, Ireland

Ryanair is the only airline that flies between the two airports, though not every day.

Unless you're willing to take some crazy indirect route from Edinburgh to continental Europe then to Shannon (and pay a lot for that) if you are flying all the way, you're going to be on Ryanair. Looks like they fly EDI-SNN nonstop 3 times/week in September (double check schedules, and be sure you read, understand, and follow all the fine print). Ryanair will be cheap, quick.

You know the old expression about quick, cheap and good? (You can't have all three, pick two).

Only rational alternative would (might) be to fly from Edinburgh to Dublin, then take the train to someplace NEAR Shannon. The train will get you close (Limerick Junction), from there you can catch a bus. The advantage of this option would be you could stop and spend some time in Dublin, if you want, plus greater comfort and a more relaxing experience en route (the train is pleasant and certainly more comfortable than flying Ryanair). Obvious downsides would be, a longer time getting there, probably greater costs, too -- depends on your needs (got large bags? good at avoiding Ryanair "gotcha" fees?), and greater complexity with multiple connection points and modes of travel.

The train probably only makes sense if you want to slip in some time in Dublin. Otherwise, if you just wanna get there and be done with it, tough it out on Ryanair (just calibrate your expectations and follow their rules).

In case the Ryanair schedule doesn't take you from Edinburgh to Shannon on the day of your choice, you could also see if there's a Glasgow-Shannon flight that day. It's fairly quick and easy to get from Edinburgh to Glasgow Airport.

or there's the boat and train combo

Regarding how to get to Ennis from Edinburgh, we checked the Glasgow/Shannon route, but it goes through London which we are trying to avoid.