You always need to make an appointment at the Apple Store, but if you are planning to use an authorized Apple service shop, an appointment is not usually needed.
You can find a list of authorized Apple service shops in your area using this link:
Find Locations- Apple
Genius Bar Reservation and Apple Support Options - Apple
Plan to drop the phone off and allow 4-6 hours for repairs. It will depend on how busy the shop is at the time. Be sure to check with the shop in advance to confirm that they have the new screen in stock. Otherwise, you will need to allow an additional 2-3 days for parts to be ordered and received.
Screen repair / replacement cost is about $300 (U.S.). If the battery also needs replacing, add another $100 (U.S.) Cost in the U.K. would be about 225 Pounds and about 75 pounds (estimated).