What is the proper way to check for null values?

What about

string y = (Session["key"] ?? "none").ToString();

You could also use as, which yields null if the conversion fails:

Session["key"] as string ?? "none"

This would return "none" even if someone stuffed an int in Session["key"].

If you're frequently doing this specifically with ToString() then you could write an extension method:

public static string NullPreservingToString(this object input)
{
    return input == null ? null : input.ToString();
}

...

string y = Session["key"].NullPreservingToString() ?? "none";

Or a method taking a default, of course:

public static string ToStringOrDefault(this object input, string defaultValue)
{
    return input == null ? defaultValue : input.ToString();
}

...

string y = Session["key"].ToStringOrDefault("none");