If the OP does decide to go with the train option, the connections are very easy - in both cases, just walking across a platform. From Changi Airport, go to Tanah Merah (there is only one way to go from the airport, and Tanah Merah will be the last stop on this train), and walk across the platform to the left, to catch a green, east-west line train towards Joo Koon. (The doors on the airport train will open on both sides upon arrival at Tanah Merah, so exiting in the correct direction is important - go out the wrong side, and you will be on the platform for trains headed away from town, not towards it.) Then this second train should be taken to City Hall, where walking across the platform puts one on a red, north-south line train to Jurong East (oddly enough, it would be easier to get to Jurong East by staying on the green line train, but since that's not where you're going, no matter). This train will go to Orchard - should be only three stops, about 5-6 minutes once on the red train. The green part will take quite a bit longer. Note that the airport to Tanah Merah train is also marked green, it seems to be considered a spur line from the east-west line, rather than a separate line.

Same deal on the way back, except you don't even have to decide which way to get off the train at Tanah Merah - there is only one way But for a 7 AM flight, I don't think you want to take the train - I think you'll get to the airport too late. So I would taxi on the way back.

There is direct access to the MRT from either Terminal 2 or 3 at Changi. From Terminal 1, it would be necessary to take airport transportation (train almost all the time, bus in the early hours of the morning) to one of the other terminals. You can also walk from Terminal 1 to 2, through the car park.

The train is very much a public-transit style system, no upgraded airport train like in Hong Kong, but at noon, the trains shouldn't be too too crowded.