Based on travel time and a guess for cost…
Madrid-Barcelona: train
Barcelona-Rome: fly. Check on the overnight ferry (to Genoa, I think) if you can’t find a reasonably priced flight
Rome-Venice: train
Venice-Budapest: fly or take overnight train ( I think there’s a direct overnight train in summer)
Budapest-Prague-Berlin: train
Berlin-Stockholm: probably fly, although it might be hard to find a cheap flight for this one
Stockholm-Amsterdam: ditto Berlin-Stockholm
Amsterdam-Bruges-London: train (EuroStar from Brussels)
London-Madrid: fly — although you’d probably be better off doing an open jaw flight and NOT backtracking to London. Use the “multi-city” on flight searches to look for flights Mexico to London and Madrid to Mexico.

For the cheapest transportation you’ll want a mix of train, cheap flights and possibly a bus or ferry. If you’re willing and able (there’s lots of hard work and some luck involved) cheapest will be to spend many hours using the tips from our “cheap flights” forum and many more hours searching out the cheapest train fares available from the individual national rail company websites. The links to the national rail company websites are in a sticky at the top of this forum and under “transportation” on the “travel tips” tab. You may end up buying a railpass for some of the longer train trips if you can’t find or purchase (some sites don’t take credit cards that weren’t issued in Europe, some don’t sell international tickets or have other glitches) a cheap train ticket. If you don’t have the time or aren’t willing to commit to nonrefundable tickets for specific dates and times several weeks in advance buy the trip planner’s recommended railpass after you’ve filled in the legs you intend to fly.

If you’re on a tight budget, re-think Stockholm and/or pick a smaller geographic area to reduce city to city transportation costs. Remember to figure baggage fees and the costs to get to/from airports when budgeting for a flight.

P.S. Your other posts indicate a tight budget. You’ve picked four of the most expensive cities in Europe (London, Stockholm, Amsterdam, Venice) and have no cities that are cheaper than average to off-set the expensive ones. You’ve got several legs that involve long and potentially expensive travel distances. This is a great trip if you can afford it.