Do I need to cut stone countertop for kitchen sink?
Does the new sink fit in the existing hole in the counter top?

If so, drop it in, caulk it, adjust (if necessary) and connect the plumbing and enjoy.
If not, you'll have to enlarge the hole to make it fit.

If you do have to enlarge the hole, you may want to seriously consider hiring someone to do the cutting for you. While you can cut a stone countertop yourself, it's not easy, most folk don't have the proper tools, and a screw up here can lead to needing to replace the entire countertop, which probably isn't a top-priority budget item for you today.
There are other questions here about cutting stone countertops, I'd recommend doing some searching to see what's involved.