The best mother daughter days out are ones that allow time to talk and interact with each other. Movies and shopping can be fun, but are often more outward focused, and probably things you do together on a more regular basis.
Start with lunch at a nice restaurant. Save the fast food for family time. You will both love the un-crowded atmosphere, and many nicer restaurants have the same dinner entrees at reduced prices. This is a great opportunity to chat about anything and everything without the typical distractions you both have in your day to day lives.
After lunch, try a new activity together. Explore both your artistic sides, and maybe discover a hidden talent! Find a local artist or community center that offer a onetime art class in painting, ceramics, or jewelry design. (Moms -book this in advance). These classes are surprisingly affordable and require no commitment. You will have a great souvenir of your day together, and a lot of fun remembering when mom spilled paint on the lady next to her. You or she may discover a new interest, and decide to make this a regular happening.
Relax, enjoy and treasure the memories of each day together.