What's the best way to store a group of constants that my program uses?

You probably could have them in a static class, with static read-only properties.

public static class Routes
{
    public static string SignUp => "signup";
}

IMO using a class full of constants is fine for constants. If they will change semi-occasionally I recommend using AppSettings in your config and the ConfigurationManager class instead.

When I have "constants" that are actually pulled in from AppSettings or similar I still will always have a "constants" class that wraps the reading from configuration manager. It's always more meaningful to have Constants.SomeModule.Setting instead of having to resort directly to ConfigurationManager.AppSettings["SomeModule/Setting"] on any place that wants to consume said setting value.

Bonus points for this setup, since SomeModule would likely be a nested class inside the Constants file, you could easily use Dependency Injection to inject either SomeModule directly into classes that depend on it. You could also even extract an interface on top of SomeModule and then create a depenedency to ISomeModuleConfiguration in your consuming code, this would then allow you to decouple the dependency to the Constants files, and even potentially make testing easier, especially if these settings come from AppSettings and you change them using config transformations because the settings are environment specific.

What I like to do is the following (but make sure to read to the end to use the proper type of constants):

internal static class ColumnKeys
{
    internal const string Date = "Date";
    internal const string Value = "Value";
    ...
}

Read this to know why const might not be what you want. Possible type of constants are:

• const fields. Do not use across assemblies (public or protected) if value might change in future because the value will be hardcoded at compile-time in those other assemblies. If you change the value, the old value will be used by the other assemblies until they are re-compiled.
• static readonly fields
• static property without set