I think most folks prefer a framework that they buy once and download because it generally offers more flexibility in customization.
Honestly the main thing that frustrates me with the very limited currently available frameworks is the total lack of updates or radio silence from developers.
Of course "out of the box" the most basic features should be ready to go- secure registration, login, log-out, account management, admin control panel, mod controls. Ability to turn registration on, off, or limit to code-based access, etc. A flexible and fully usable admin control panel is a must. Should be able to turn features on and off, modify pages, users, etc with ease.
As for actual pet-game features it depends highly on what style of game you want to cater to. The more adaptable you make it, the more people will be able to use it. Making features togglable is a great way to offer something for everyone.
Take a look at most modern virtual pet games and figure out what they have in common. I'd recommend starting small and simple, and then listen to community feedback.