What's the best way to go about potty training in public places? I bring her potty seat along, but she won't use it when we're out of the house.
Continue to take the potty with you. There's no need to put her back in diapers. Since she might have an accident away from home, be prepared to clean her up by taking along an extra set of clothes.
New places in general and public restrooms in particular are difficult for children to successfully navigate. It all takes time, but she will eventually learn to use them but it might be another three months until she does so. Check out this article for more help: 5 Public Potty Pointers.