The first thing you need to do is to remove old tiles. Cut out the grout using a grinder to easily remove the old tiles. Thoroughly clean the area from which you removed the tile.
Image Courtesy: hertfordshire.tiledoctor.co.uk
Using the old grout lines as a reference lay the new tiles. Place the tiles on the mortar with a twisted motion and tap it into place using a mallet. To make sure that the layer of mortar you had applied earlier is thick enough to cover the bottom of the tile completely, pull up only the first tile and see its back. Apply more mortar and comb it with the notched trowel if necessary. Give enough time to the mortar to set properly before moving on to the next step.
Now you need to fill the grout lines. Apply grout with a grout float and firmly push the freshly applied grout into the grout lines. Wipe off the excess grout using a damp sponge. Wipe gently to avoid removing grout from within the lines. Continue wiping the tile until all the excess grout has been cleared and the tile edges are clean as possible.
Image Courtesy: todayshomeowner.com
You may have to wait for a whole day to let the grout dry completely. After it dries, apply a layer of grout sealer.
Now it comes to installing the tile floor transition. Using a miter saw cut the prefinished flooring transition moulding. Size it carefully so that it can fit the length between corners. Use a drill to make pilot holes in the moulding and hammer finish nails in these holes. When you are done, you have successfully installed a tile floor transition.