partial update via javascript

What you need is AJAX. If you look at docs it says

JavaScript can also make requests to the server, and parse the response. It also has the ability to update information on the page. Combining these two powers, a JavaScript writer can make a web page that can update just parts of itself, without needing to get the full page data from the server. This is a powerful technique that we call Ajax.

For this to work you first need to change your form to have remote: true option which basically allows your forms values to be submitted by js. Your form will look like

<%= form_for @todo_list, remote: true do |f| %>
  ...
  // form fields
<% end %>

<div id= "some-id"> // to target this div to render your partial from js
  <%= render 'show'  %>  // it doesn't make sense to have your partial inside form and 
</div>

it should be outside your form

In your create action you need to have a respond_to block

def create
  @todo = Todo.new(todo_params)
  respond_to do |format| 
    if @todo.save
      format.js{}  # this will allow rails to look for create.js.erb in your views/controller directory where you can use your js to render your partial or append newly created todo
    else
      format.html{render "new"}
    end
  end
end

In your create.js.erb template use your js to render partial or append your newly created item

$("#some-id").html("<%=j render "show" %>");

OR

Instead of rendering whole partial again what you can do it just append your newly created todo in your table by js

$("#table-id").append("<%=j render partial: "todo", locals: {f: @todo} %>");

Here your _todo.html.erb will contain

<tr>
  <td><%=f.name %></td>
  <td><%=f.description%></td>
  <td><%=f.is_public  %></td>
</tr>

For details checkout Working with javascript in rails

UPDATE

If you look at your partial you have

<table id="list" border="1" >
  <% @todo_lists.each do |f|%> // your @todo_lists is a collection of todos 
    <tr>
      <td><%=f.name %></td>
      <td><%=f.description%></td>
      <td><%= link_to new_task_path %></td> // you don't need to specify method here link_to by default uses a get request
      <td> <%= link_to 'Destroy', f, confirm: 'Are you sure?', method: :delete%></td>
    </tr>
  <% end %>
</table>

Change your controller method to this:

def create
  @todo_list = ToDoList.new(params[:to_do_list])
  @todo_list.member_id = current_member.id
  respond_to do |format|
    if @todo_list.save
      @todo_lists = ToDoList.all // get a collection of todos to render in your partial
      format.js{}
      format.html { redirect_to @todo_list, notice: 'Lista ' +@todo_list.name+ ' foi criada com sucess!' }
      format.json { render json: @todo_list, status: :created, location: @todo_list }
    else
      format.html { render action: "new" }
      format.json { render json: @todo_list.errors, status: :unprocessable_entity }
    end
  end
end

And in your create.js.erb file

$("#show_list").html('<%=j render partial: "to_do_lists/show", locals: { todo_lists: @todo_lists } %>'); //You don't have an element with id = todos. You have it with show_lists so change that

Also render your partial after closing your form. It doesn't make sense to show todos inside a form