This would be correct:
Danke schön, gleichfalls!
The word »Appetit« ("appetite" in english) is a noun. In German all nouns must always be written with an uppercase first letter, so »appetit« is wrong. Only »Appetit« is correct.
»Guten Appetit« is an ellipsis. The whole sentence is one of those:
Ich wünsche dir einen guten Appetit.
Ich wünsche euch einen guten Appetit.
Ich wünsche Ihnen einen guten Appetit.
All three sentences literally are in English:
I wish you a good appetite.
Have a nice meal; Enjoy your meal.
The personal pronoun (»dir«, »euch« or »Ihnen«) depends on the relationship that you have to the persons you are talking to. You use »dir« when talking to one child or to one friend, and »euch« if you are talking to two or more children or friends. If you are talking to adult people who are not your friends (i.e. customers, your bosses boss, strangers) you use »Ihnen« (both: singular and plural). But you don't need to worry about it when you use the very common ellipsis »Guten Appetit«.
(see: T–V distinction in Wikipedia)
Synonyms for »GutenAppetit« are:
- Lass es dir schmecken!
- Wohl bekomm's!
There are also lots of regional variations.
This is the standard expression for situations where you would say
Rule of thumb: »danke« is equivalent to »thanks«. »Danke schön« or »danke vielmals« are equivalent to »thank you« or »thank you very much«. So »danke schön« is more formal than just »danke«.
This is an adverb. The literal one-word-translation is:
But this is not very useful because in English you construct phrases where you would use »gleichfalls« in German in a different way. Here, in this context (after »danke« or similar expressions) you better translate it as:
the same to you
Even better: use a different way to say, that you wish your conversation partner the same that he/she wished to you.
You can use »gleichfalls« not only in friendly situations. You can use it also for example when someone insults you:
Leck mich am Arsch!
»Leck mich am Arsch« is literally »lick me at my ass«, but you don't say that in English. »Fuck you« is »Fick dich« in German, and can be used in German since about 15 or 20 years.
Lets come back to your original dialog.
The complete dialog would be in English:
Enjoy your meal!
Thank you, enjoy yours too!