First: Trim

Remove the trim and baseboards in the room that you will be working in.  If you are careful as you are doing this, then you will not have to buy new trim work when you are done.  You will just be able to put back what you have taken off.  Take any cabinets, appliances, or anything else out of the room.  Even if it is built in.  Take the time to cover all vents with removable painters tape and plastic.  This will prevent the dust from getting into your ventilation system.

Second: Tile

Use a hammer on the first tile to get it broken as in image 1.  Next, use a chisel to remove the pieces of the first tile that are still stuck.  After you have removed the first tile, you will then use your chisel to begin removing the other tiles around it.  This is done by putting the chisel at the bottom edge of the the next tile and then hitting the end of the chisel with a hammer to break the tile loose.  They should come up pretty easily as shown in image 2.

Third: Underlayment

After you have all the tiles up, you can then start removing the underlayment for the tiles.  Taking your screw gun, take out all the screws that are holding the underlayment on the floor.  Use a flat shovel and a roofing rake in order to pry the edges of the underlayment away from the floor.

Fourth: Subfloor

After you have gotten up all the underlayment out of the way, you will take the shovel and use the flat edge of it in order to get rid of any adhesive that is left.  For larger areas use a carpet scraper, and for smaller ones you can use a scraper or your hammer and chisel.  Clean any leftover screws or adhesive with a shop vacuum or a broom.  Make sure that all the wooden subflooring is completely level before you put your new flooring in.