Ok... im talking from personal experience here.
London to Paris: 45min plane ride from Heathrow to Charles De Gaulle. That plane ticket was $75 with Air France though. Plus you have to include going thru security and getting your luggage, plus CDG isnt smack dab in the middle of Paris.
Or you could take the chunnel. That was about $75 depending on what time you travel and your age and what class your wanting to take. Takes about 3hrs and you leave Waterloo station and Arrive right in the middle of Paris at Gard De Lyon or Gard to Nord. Or you can take the ferry that leaves from Dover and takes you to Calias, France, or something like that. Never took the Ferry, so I dont know how much it costs, but if you have a railpass i think that is included at either a discount or free. That would take the longest.
London to Dublin: Flight is like an hour or so. Cheap rate is using either Ryan Air or Easyjet. You can probably find a cheap rate costing you as cheap as $50 or less depending on many things. Or you can buy a railpass for the UK and do it that way, but will have to take a ferry to get to Dublin. Dublin to Paris, quickest is by plane. If you have a certain timeframe, I wouldnt want to waste my time taking a ferry, train, then ferry/train to Paris. Waste a lot of travel time doing it that way. Plus, travelling by plane is VERY cheap!!! Like I said, RYANAIR or EASYJET.
I would fly into Dublin, then take the ferry/train to London, and then take either the plane or chunnel to Paris if those are the places you are going to. Or you can go the other direction and start in Paris. See which is cheaper to fly into. But I believe London might be the cheapest to fly into first.
Hope that helped!