If the strainer basket still has cross-shaped bars at the bottom, a special wrench is avaialble for less than $10 at any hardware store to try and unscrew it. If it is old and the bars snap off, an internal pipe gripper is available from a plumbing supply house to turn it out.
To install the new on: You will reach into the hole with a toothpick or screwdriver and remove a rubber gasket which is sandwiched between the bottom side of the tub and the face of the drain elbow. A new gasket will then be slipped in. A fat bead of putty is then placed on the bottom of the new strainer basket and it is screwed into place. I say a fat bead, but you do want to keep putty off the threads, and try not to use so much putty that is is forced down thru the hole.
The are 2 common sizes of basket threads, and probably a dozen less common sizes. Take the old one with you to a plumbing supply to obtain a replacement.