That's fine you want to remove the old flooring and adhesive but what's under that?!! Is it the sub-floor or some type of 'underlayment' like Lauan? (See the 2nd-to-last section about Lauan.) If so, manually rip out the lauan. If it's the sub-floor, directly, that you're removing the existing tile from, it might not be too much fun to clean. I'd check the warranty of the flooring first to see if it requires some type of underlayment. Lauan, for example, is no more than 1/4" thick. If you don't use it (again, you didn't say enough for me to be sure), you may void your warranty and you may have a problems with the flooring over time. If the sub-floor is sheet plywood, then you may be OK with the install but you may still have problems. (I was initially referring to tongue-and-groove sub-flooring.) You have to ask yourself, "What may happen in either case if you don't let the sub-floor dry properly? Suppose there's some 'residue' left behind and the glue won't stick any more after a year?" You see what I'm saying?
Do some checking first. Oh, if you do put down the lauan, you MAY have to cut your door moldings a little. There is s special saw for this and you can do it yourself. Also, the 'transition' into other rooms will be 1/4" higher if you use it. I could be way off on this but maybe others will agree/disagree.