What's the most cost effective way to get from Ireland to England
Your exact departure and arrival points might influence your strategy, but we flew from Shannon Airport in western Ireland to Heathrow Airport outside of London on Aer Lingus. Quick flight, right price. Other airlines may offer flights on a different time schedule or at a different price, but might not serve the same airports you have in mind.
Ferries across the water are another option, but consider their cost and travel timeframe.
See also a brief discussion of ferries vs flying at http://www.ricksteves.com/travel-tips/transportation/trains/ireland-rail-passes.
Most popular option would be to fly... with so many daily departures from Dublin, Shannon, Cork, Knock, Belfast. Relatively good value fares available too. Check Ryanair, Aer Lingus, Cityjet, Flybe.
There is the ferry from Dublin to Hollyhead, Rosslare to Pembroke, Belfast to Cairnryan / Liverpool. There is also the Eurolines bus.
You might find that a fare for a foot passenger on the ferry or on the Eurolines bus is cheaper, but I would strongly recommend flying. Not easy to get to/ from some ports, and the Eurolines bus is a lengthy journey.