Take a look at that wood strip from underneath if you can.
Usually, a tiled counter is tile over plywood and that wood trim its nailed to the plywood base but sometimes they come as a complete prefabed unit and the base and trim are made of particle board with a laminate over the exposed edge.
If the trim is nailed on I'd take it off, replace the countertop and then put it back on. That leaves less chance of mucking up the piece you want to keep. (You'll probably find it faster to just remove the counter entirely and put down new plywood!)
Your sink should just have a few clips on the underside. You can lossen those up, disconnect the drain and water feeds and pop that right out. It goes right back in the same way once you've resurfaced the counter.
P.S. - Tile counters look nice when they go in but they usually end up looking like crap after a few years. To many gaps to catch junk and the grout stains easily. I'm not a fan of them.
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