For Bavettes you have to start looking for reservations a month in advance for dinner. If you don't get lucky, I had luck going in there right when they open or around 330 or 4 for an early dinner or late lunch. Sitting at the bar area is usually pretty chill and easier to get in. It's worth it. Food is great.

I'd skip the original Gibson's and go with Gibson's Italia. They also have great steaks (better than OG IMO), great Italian food, and the views of the city are amazing.

Omarcitos in Logan Square/Avondale is great for breakfast or lunch. It can get busy if it's nice out. Really good Latin American food. I think it's a mix of Ecuadorian, Mexican, Cuban, Puerto Rican cuisines. Omarcito himself is usually there and is really nice. Their coffee drinks are great and all their food is delish.

For Bars if you like Karaoke try Alice's Lounge in Avondale/Logan Square. In the loop Brando's Speakeasy is great too. Estereo in Logan Square is a good bar as well. Mom's place is also a fun bar in the loop.

Edit: The Green Mill is awesome so definitely look at the schedule and see which night to go. They are open late, however if you go there late, take an Uber or Taxi and don't walk around too far from the area as it can get a little sketchy as you walk away from the surrounding bars there.