When I log in to a website for the first time, 1Password does not prompt to save my login details
There are a number of reasons this could be. First, be sure that "Ask to save new Logins" is checked in the Logins section of the browser extension's settings:
There are other common reasons a login might not be saved that could be causing you trouble. We're working hard to try to get 1Password saving more and more sites automatically, but we haven't convinced the entire internet to stick to standards. Yet. 
In this case, we recommend saving a Login manually. Here's how:
Navigate to the web page in question using your web browser of choice.
Enter your username, password, and any other required fields if applicable (security code, PIN, etc), but DO NOT submit the form.
Click the 1Password button in your browser's toolbar, and select the "+" button in the upper right hand corner:
Change the title and make any notes (if desired):
Click the Save button in the upper right hand corner.
Now your login is saved in 1Password, just a click away!