Come: kimasu (polite), or kuru (informal)
Eat: tabemasu (polite), or taberu (informal)
taberu (to eat)
kuru 来る
By virtue of the fact that Japanese food is inherently Japanese, it would be fair to say that the people of Japan eat the most Japanese food.
The traditional phrase that the Japanese say before they eat is "Itadakimasu," which is a polite way of expressing gratitude for the meal.
Japanese usually say "gochisosama" (which means "Thank you for the meal") after they eat.
It's "Itadakimasu".
Asa gohan o tabemasu
Oustreria kara desu
Ofurosato ha dochira deshouka?
it's very similar to *Listen, which is Kiite but, to come is Kite
'Itadakimasu' is said before partaking. For a more conversational "I'm planning to eat ... (soon/later/so will you come with me/etc)," you may say 'Nanika tabeyou to omotte imasu ga' ("I'm thinking about eating, so ...")
できるだけ食べてね (dekirudake tabete ne) may be translated as 'eat all you can' or 'eat as much as possible' in Japanese.