Mike, first of all, you need to determine if your old pickguard is under or over the finish. If it's under the finish, I can't help you any further, as my experience is solely with over-the-finish PGs If it's under the finish, I'd recommend you check out Frank Ford's article which gillymor linked to above. I'm pretty certain it will be an under-the-finish pickguard, these were the standard until the mid-1980's when CFM started to finish the top prior to sticking the PG over the finish. I repeat, I'm almost certain that yours will be this type of PG, so you need to seek guidance elsewhere.

If, by some strange chance, it's over the finish - you can remove the old guard by carefully warming it using a hair-dryer. Start at the pointed end and when it's warm, gently lift the point with a fingernail, and insert a spatula. Continue warming along the PG, and gently ease the PG away from the top with your spatula, a little at a time. Eventually you will be able to lift the whole PG away. Remember - you're warming the PG, not cooking it!

Any glue remaining on the top can be removed with naphtha.

Here is a good video showing the technique, made by the superb luthier, John Hall of Blues Creek Guitars.

If you purchase a new PG, it should already have adhesive sheet on the back, and you just peel away the backing-paper covering.

The easy way to get a new PG placed perfectly is, before peeling the backing-paper away from the guard, place it on the top and line it up exactly where it needs to be, then use two or three small 'hinges' made from masking-tape or electrician's tape along the back edge to temporarily hold it in place. Lift the free edge up and let it drop down two or three times to check that it drops in the exact right place, then lift it up and very carefully, so as not to disturb the hinges and placement, peel away the backing-paper. Let the guard drop into place - if it's slightly 'off-line' it's usually possible to slide it a small amount, before pressing it down hard, to adjust. When you're happy, use a soft duster and gently press the PG down to get full adhesion.

Here's a nice video demonstrating the above.

That's how I've done the job several times without any problems. However, I don't know what your skills are, I'm not a luthier, I'm an enthusiastic amateur, and I don't accept any responsibility if things go wrong for you! I strongly recommend that you cast around YouTube for other informative and instructional materials and, if you have any doubts at all, take the instrument to a competent luthier to have the work done!

Good Luck!