I remember my first invisible zipper replacement - I was nervous like you would not believe.  But what I did was unpick one side only.  The left side facing you is the best to do, then insert the zip using the other side as the guide.

Always insert an invisible zipper beginning on the left side with the outer facing you.  Make sure that you have a little bit of room from the top of the garment to the top of the zip.  Most zips have a small plastic or metal lock at the top.  This lock should be positioned about 3mm (1/8 in) below the top.  Push the teeth across with your finger and sew down right next to the teeth.  Close the zip and pin the opposite side in place making sure that the bottom and the top are even.  Open the zip and begin to sew from the bottom of the zip up to the top.  Close the zip and begin to sew at the seam below the zip, up to the zip and BESIDE the teeth just up to and past where it is joined to the fabric.  This stitching must be close to the zip and the stitching must be straight otherwise it will not look right on the right side of the fabric.

In the past I have had some clients ask why I have not sewn to the bottom of the zip.  With an invisible zip, it is not possible to sew to the very end.

You can buy a kit that is called an Invisible Zipper foot attachment.  There are a few in the kit because they are designed to cater to all the different types of sewing machines, but I prefer to use my normal zipper foot that comes with the domestic sewing machine.

Happy altering
Judith aka genie