There are a metric crapton of places to eat in Conway. See Restaurants for the complete list. This is a suggestion for places at different price levels. All listed entries have alcohol.
$$$$ - Oak Street Bistro has closed. If you loved their white-cheddar dill soup, Pasta Grill sells the real deal and has a similar price range.
$$$ - Mike's Place - Louisianan (ish). About $12 / plate
Both places are great for dates, but if Pasta Grill is busy, you may have trouble hearing each other.
$$ - JJ's Grill or Zaza's - Burgers or Pizza. $10 / person
$ - List is almost too long to make a call. See Cheap Food for a complete list. Well known cheap eats include Stoby's, Bob's Grill, and Holly's Country Cooking.