36 Elliot Street Milton
I think that's where Kate's used to be? I didn't know there was a restaurant there anymore.
Milton has very few restaurants. For breakfast, Newcomb Farms and A Lighter Fare are both good options. For lunch or dinner, I'd head into Quincy, since there are lots of good restaurants there, including Pho Yuen Dong (Vietnamese), Tullio's (Italian), and Alba (New American/Mediterranean).
I moved to Milton last year, the best bet for the area you describe is probably a short ride up Dorchester ave. to the Ashmont Grill (across from Ashmont T Station). I live near Eliot St. and Kates' is no more. I hear Alba's is closing soon (?) so that would be the next best offer, but in the closing mode, who knows what to expect... Unfortunately, you have joined me in a neighborhood culinary wasteland by proxy.
I used to love going to Java Joe's on a Sunday afternoon and get a coffee and a tuna roll up for lunch. They have a funky, slacker vibe there. I havent been in a few years, so I dont know if it's changed at all, but it used to be a good choice for sandwiches and salads.
I 2d the vote for a Lighter Fare - above the Fruit Market place in East Milton. Great breakfasts there.
After that I'd say head to Quincy Center for more dining options, as mentioned above.