Today I tackle one of the trickiest alterations I do, shortening a vented sleeve on a suit coat. In order to be concise, I can't give you a step by step process here, but I'll hit the highlights. My goal is to get a tutorial on my website as soon as possible.
Taking out the existing stitching will take a little time and care. Remove the buttons and put them inside a pocket, pinning it shut for safe keeping. I always do one sleeve completely, then the other. That way I have a reference if something goes wrong. Of course digital photographs are great for reference too.
I detach the lining and rip the stitching on either side of the vent. Usually one side is mitred, the other not. Once the sleeve is opened up, I can mark the new hemline. Press this into place. At this point you can shorten the lining too, but if it's an inch or less, I leave it alone. The extra ease in the sleeve in OK.
The mitred corner on the outer side of the vent (the side where the buttons were attached) is difficult to duplicate. You will get it with practice. I fold the hem allowance along a diagonal line, right at the fold line of the newly pressed hemline. This is the second photo above. This will give you the ability to make two more folds on top of this diagonal fold, thus mitring the corner. Press firmly.

Now I unfold this, turn it right sides together and stitch along the fold I just pressed. When I first started out this was a lot of trial and error, because once you stitch it and turn it, sometimes it is uneven. It helps to mark the apex of your foldlines with a straight pin

The other side is much easier. You just fold right sides together and stitch parallel to the raw edge. Flip and press, making sure the two sides are now even.
Lastly, I replace the buttons. If you do this before attaching the lining, the stitches will not show. It adds professionalism to the garment.
I leave a little movement ease when attaching the lining. I pin it in place, then draw back a scant ½" or so, before I sew. If you look at ready wear, you'll see what I mean. It keeps your sleeve from buckling and looking bumpy. Here is a photo:

This is definitely one of my most time consuming alterations, and you can see why. Some days it goes in on the first try, but most days I have to redo that mitre several times. Don't get discouraged if you are trying this for the first time. Go to Good Will and get a jacket with which to practice. It does get easier, just not faster! But it also gives me options when shopping for my husband or sons. I can buy a suit that is on clearance, knowing I can alter it to fit.