Yep that's right, after coolant change and before I did the thermostat it started running a bit cooler.
But I think it didn't take any longer to warm up - if I understand this thermostat correctly, I would say when the car got cold (ie overnight) it shut completely and until it got the signal to open it was closed - hence the car warmed up quickly. That's also why the needle went all the way up after you started a cold car. Then, after it opened for the first time, it couldn't move properly to keep the engine at the correct temperature - I'm guessing that big spring inside the thermostat gets worn after time.
As far as the actually changing it, it's not hard to do but it's extremely fiddlish. The main bolts are hidden so you have to feel them up like a girl in the dark, and there's also very little room for tools - but I did it and I'm far from a mechanic, so it shouldn't be a problem for you. You need to take out the battery, battery tray and the computer. And don't forget to buy enough jubilee clips (watch out for sizes as every hose is different in diameter I think).
After I changed the thermostat I opened the radiator bleed valve and the one on the hard pipe infront of the engine and poured in coolant until it came from the holes. Then I topped up the coolant and started the car without the expansion tank cap. Waited until the thermostat opened and then set the heating to highest setting so the coolant went into those hoses also.
Closed the expansion tank cap, idled it some more, waited for the car to cool halfway and opened the expansion tank again, for any remaining air to escape.
Then I went for a drive and opened the expansion tank again with car half warm. Added just a little coolant next morning when the car was cold - I hope that got all the air out.