Depending on the age of the linoleum (not to be confused with vinyl flooring), it may contain asbestos. IF the linoleum was installed before 1980, there's a VERY good chance that it contains asbestos along w/the glue. Now, what to do and what NOT to do.
DON'T use any saws, power or hand, to cut the stuff up. There is no 'direct' hazard as long as you don't create any dust. Sooooo, when you pull it up, try to keep it in tact and in in reasonably-sized pieces until you get it outside. Once it's outside, you can carefully bend and fold it into the proper 'size' that the trash company will take away at the curb. (That is, if you don't have a dumpster ...)
I suggest you 'take up' the flooring along with whatever 1/4" underlayment there is (assuming there is an underlayemnt.) If there's hardwood or concrete under there, well, you can pretty much forget about trying to remove the glue. For hardwood, you'll need a LOT of sandpaper. Lauan (pronounced LOU-on) is an underlayment that's easy to handle. If your underlayment is 1/4" plywood, it's gonna be tougher to handle. It may come up in sheets as big as 4' x 8'. Just find the seams and go from there.
Now, IF your flooring is plain, old modern vinyl flooring (which a lot of folks refer to as Linoleum from the 'old days'), you needn't worry about asbestos. For me, the trick is the same as with removing linoleum. Look for the seams and get a pry-bar or claw hammer and start ripping it up. You can use a Utility Knife to cut the slice the material on BOTH sides of the seam (so you can peel the material OFF the floor) to find a seam. Kinda like peeling a banana. You might have to remove the trim/molding all around the perimeter of the room as well as under the cabinets to do this job. Just be careful not to damage the cabinets and walls. (And be careful removing the trim too unless you're gonna replace that too.) I suggest you have your floor installed W/O the trim in place. That way, the trim can hold the flooring 'down', so to speak.
Yes, there are other ways to do this job which some of the other 'guys' will probably post up. So stay tuned! My best to ya and hope this helps.