- Remove the 4 bolts on the front of the old stem to release the handlebars. (I generally let them hang from the brake and gear cables - not sure what other people do.)
- Loosen the 2 bolts on the clamp that attaches the stem to the steerer tube.
- Remove the bolt in the top of the steerer tube and take the top cap off.
- Slide the old stem off the top of the steerer tube, slide the new stem on.
- Replace the top cap and do up the bolt firmly, but not too tight.
- Align the stem with the front wheel, and tighten the bolts that attach it to the steerer tube.
- Attach the handlebars to the front of the new stem.
The tricky part is getting the bolt on the top of the steerer (which sets the compression on the headset bearings) just tight enough, but not too tight. I generally do this by overtightening slightly at first - you know you've overtightened when you lift the front of the bike by the frame and the front wheel
doesn't swing freely to one side. Once I'm at that point, I undo the stem bolts - very important step - loosen the top bolt by 1/8 of a turn, tighten the stem bolts and do the check on the front wheel again. I repeat this until the front wheel swings to the side freely when I lift the front of the frame, which means the top bolt is as tight as it can be without being over-tight.
The other thing you might find is that your new stem has a different stack height to the old one. (This is the measurement from top to bottom of the clamp that attaches to the steerer tube.) I'd measure the stack height of your old stem before you go. If the new one is different, you'll probably need to buy a spacer or two to make it fit. The most common spacer sizes are 10mm, 5mm and 3mm, although you can get multiples of these sizes, such as 15mm, and it looks like you've currently got 3 10mm spacers on your bike.
When you take the top cap off, you'll notice that the steerer tube is a few millimetres lower than the top of the stem clamp. This is normal and correct, so you want to keep it the same when you put the new stem on.