I am putting slate mosaic tile in my entryway from my garage (it used to be vinyl type). The entryway opens into by basement room via a doorway that is approximately 32 inches wide and the basement room has carpet. The floor is concrete and there is currently a metal transition strip in place. I want to remove the metal transition strip and just butt the carpet to the tile. Following the advice of my local HD guy, I bought a tack strip that just has little concete nails to go down into the concrete and then a carpet tack strip on the top side. How do I go about accomplishing this project? Do I lay the tack strip first, remove the old transition first, tile first..

You will need an edging strip for the slate tiles. Shown below, comes in mill finish or gold. Otherwise the first row of tiles are subject to breakage and damage. Since you have a concrete floor you could just install it with construction adhesive but it would be much better to drill two or three holes for each strip and use plastic anchors and flat head screws to install.
Install the edging material, then lay your tile. Install two rows of carpet tackless strips about 3/4" from tile edging using construction adhesive. Wait about a week for the construction adhesive under the tackless strips to cure then kick the carpet in, butting it up to the tile edging strips.

I agree with Harold about the edging strip for tile.

Before installing tackless strips clean concrete to remove dust.
Apply 2 coats contact cement to concrete and tack strips. Install first strip about 3/8 inch from edging strip. U can use your index finger as spacing guide. Tap down with hammer to securely set on contact cement. Drive concrete pins on tackless into floor. Stretch carpet onto tackless and cut 1/4 inch longer, onto tile. Push this 1/4 inch down between edging and tack strip with a stiff putty knife.

Chuck

Awesome! Thanks for the picture - I know exactly what your talking about now.