While user stories should have good acceptance criteria, checking that these criteria are met is really just the starting point before engaging in deeper testing.
Without detailed test cases, it can be difficult to work out what to test outside of the acceptance criteria. Using test ideas and heuristics to come up with these ideas are important skills for good exploratory testing.
Taking an exploratory approach to testing gives testers more freedom to choose what to test and how to test, but they can then find it hard to decide what to test.
It's helpful to think in terms of test ideas.
Test ideas are brief notions about something that could be tested - Rikard Edgren
Test idea: an idea for testing something - James Bach
When tasked with testing something new, you don't necessarily know how to unearth interesting test ideas and so following rules probably doesn't help.
Under such conditions of uncertainty (which are normal in software development), look for methods or ways of coming up with test ideas that might work, while acknowledging that they might not - these are heuristics.
A heuristic is a fallible method for solving a problem or making a decision.
A heuristic is an experience-based technique for problem solving, learning and discovery.
A heuristic is a way to help me come up with test ideas - Lee Hawkins
You'll likely build up your own toolbox of heuristics to draw from as you become more familiar with them and realise their power.
To get started with the use of heuristics, try some consistency heuristics. A great example comes from Michael Bolton in the form of HICCUPPS, which is a mnemonic as follows:
As an example of using the "Product" part of this heuristic, note that in Trello, there is an inconsistency in the way fields on cards can be edited. Clicking in the Description or Tasks fields puts the field straight into Edit mode, whereas to edit a Comment requires an explicit click on the "Edit" action. This is a case of the Trello product being inconsistent with itself in terms of editing fields on a card.