Hi, this is Gary; thank you for using Justanswer.
On the floor, you can apply an approved underlayment material right onto the existing finish flooring. This can be an underlayment grade plywood, or a cementitious tile underlayment. You will have to follow the manufacturer's instructions; both for the tile, and for the underlayment.
For installation over linoleum, an underlayment rated plywood is better, as it does not require a thin set mortar bed for installation, simple construction adhesive will work with nails or screws following the nailing pattern marked on the board.
If your flooring is installed on an underlayment now, rather than on top of the subfloor (you can tell by looking into a floor register to see how many layers of wood are present, you may have to pry one side of the trim away from the opening), you can pull up the flooring and underlayment together, and start fresh. Most Builders do not pay to have the underlayment glued down, so you will just have to pull or pound down any nails or staples that did not come up with the flooring.
As far as Counter Tops go, the tile will not adhere to the laminate counter material; it must be removed. A double layer counter sub-base should be constructed, using one layer of at least 5/8" thick Exterior Plywood, and one layer of 1/2' cement board. For the counter edges, if edge trim is not available that will "wrap" the front of the counter and the exposed edge, glue and nail a piece of the plywood onto the front and use a regular mastic designed to hold heavy tiles to apply the nose tiles to the front of the counter.
Quarry tiles are best set on a floor using thinset cement; on countertops, I have been successful using either mastic or thinset.
To remain sanitary, the tiles on the countertop must be sealed often, with a food-safe sealer; ask about this at the tile supply house to be certain that the tiles you have chosen will even be able to be safely installed on a counter where food is prepared.
If you have any additional questions, feel free to ask.
Good luck,
Gary
Building Inspector
ICC Member, IAEI Member, ICC Multi-Certified Inspector; Licensed Electrical Contractor, Instructor