Even inexpensive vinyl tile can brighten a room. For those good looks to last, though, it's better to move up a grade or two. You'll pay more for VCT that is thicker, realistically embossed and coated with a tough wear layer on top, but a sturdy, durable tile will install more easily, look better and last longer. Additionally, vinyl tile that is too thin has a tendency to "telegraph" the surface underneath, eventually showing every seam, chip and defect.
Substrate is the term used to describe the surface onto which you install the vinyl tile. The condition of a substrate makes a big difference in how well the new tile will adhere, look and last. If the existing linoleum is a lumpy, torn-up mess, don't even consider using VCT without either tearing up the linoleum or covering it with a new underlayment such as 1/4-inch plywood. The linoleum should be free of wax and grease, intact and well-adhered to the subfloor. Heavily embossed linoleum might require using a leveling compound first; consult the manufacturer's installation instructions.
A good quality vinyl composition tile can often be installed over existing linoleum that is clean and smooth or that has been treated with a leveling compound. Should the condition of the substrate warrant it -- or the tile manufacturer recommends it -- use a high quality adhesive made specifically for vinyl tile. Apply the adhesive following the directions on the package label, paying particular attention to the recommended trowel notch size. Too small a notch and the tile might not adhere well, too large and the job will use more adhesive and be a lot messier than necessary.
The adhesive on peel-and-stick tile is unforgiving. You probably won't be able to pull a tile up without destroying it, and it won't stick as well the next time. You are going to be looking at your new floor for a long time, so don't rush the job. Decide where to start and where to finish. Figure out what cuts you will need to make and how to accomplish them. Once all the tiles are down, it is recommended to use a 100-pound roller on some grades of vinyl composition tile. You can rent one at most home improvement centers.