I think both options would work for your particular application.
I just re-did my floors (1st timer fyi) and they turned out perfect. I took out the tile, resurfaced the floor and added a buttload (17-20 bags) of self leveling concrete (forgot the brand but it was about $30/bag) just to get it level.
You will maybe hear about getting your floor "flat" instead of worrying about getting it "level" - I personally subscribe to both philosophies b/c I didn't want people walking sideways in my dining room.
Good luck and make sure you budget time beforehand. I had to go back and forth to Home Depot a gazillion times for supplies so be sure to plan ahead of time.