How to access Winform textbox control from another class?

I would recommend that you don’t. Do you really want to have a class that is dependent on how the text editing is implemented in the form, or do you want a mechanism allowing you to get and set the text?

I would suggest the latter. So in your form, create a property that wraps the Text property of the TextBox control in question:

public string FirstName
{
    get { return firstNameTextBox.Text; }
    set { firstNameTextBox.Text = value; }
}

Next, create some mechanism through which you class can get a reference to the form (through the contructor for instance). Then that class can use the property to access and modify the text:

class SomeClass
{
    private readonly YourFormClass form;
    public SomeClass(YourFormClass form)
    {
        this.form = form;
    }

    private void SomeMethodDoingStuffWithText()
    {
        string firstName = form.FirstName;
        form.FirstName = "some name";
    }
}

An even better solution would be to define the possible interactions in an interface, and let that interface be the contract between your form and the other class. That way the class is completely decoupled from the form, and can use anyting implementing the interface (which opens the door for far easier testing):

interface IYourForm
{
    string FirstName { get; set; }
}

In your form class:

class YourFormClass : Form, IYourForm
{
    // lots of other code here

    public string FirstName
    {
        get { return firstNameTextBox.Text; }
        set { firstNameTextBox.Text = value; }
    }
}

…and the class:

class SomeClass
{
    private readonly IYourForm form;
    public SomeClass(IYourForm form)
    {
        this.form = form;
    }

    // and so on

}