Nice Restaurant with Infant [DFW]
I agree with the advice and recommendations of DalaiMama, especially with the advice to go early -- arriving between 5:30 - 6 pm will allow you to get in and out. I mostly have East Dallas recs too -- the dearly departed Kitchen 1924 was great with kids -- probably the closest you could get to real foodie-food and still be kid-friendly. We have done well at Angelo's, Highland Park Cafeteria, and Big Shucks. Be ready to ask for the check as soon as the entrees arrive, and get your dessert to go. You may check out this new website mentioned on this board and on the Dallas News Eats Blog --
Finally, I would encourage you to disregard the comments of hellpaso -- you sound respectful of your fellow diners, and you definitely do not have to eat McDonald's until your baby attends Charm School. See you at 5:30!
I am going to assume that you are not talking about very fine dining (Stephen Pyles - that kind of thing). Once you get to that level, I think the standard is a bit different.
You can always call a place and ask if you can bring a baby. I've done this before when traveling. The reservationist will usually be pretty frank.
You don't specify where you are in Dallas, so I will give East Dallas/Uptown recs.
My first piece of advice is go early. I have dined out at various reasonably nice places but made sure to go around 6pm. I think that is respectful of other patrons. Places tend to be less busy at that time of day as well.
Teppo is great for sushi and they are fine with children. It's our six-year-old's favorite restaurant ;>
Amore in Snider Plaza is good Italian.
Taverna on Knox is slightly more upscale Italian. I would not take a baby there on the weekend because it gets extremely busy and extremely loud.
Cafe Madrid is good for tapas.
The Grape has a very family friendly Sunday brunch with nice food.
Shinsei is Asian and they are somewhat kid friendly.
We also like Bisto N at Nordstrom in Northpark Mall. Granted, it is a chain, but it's a nice place to eat dinner, they have a full bar, the food is always good - a bit pricey, but good; and they have no problem at all with children.
La Duni is also great with small children.