Activity diagram is one of the those with widest abstraction range in UML. An activity can be used for anything between a business process (very abstract, comparing with software system) and a single method algorithm (code-level, practically blue-print, meaning kind of abstraction ground level).
Use Cases on the other side are in practice very limited in their abstraction. They show the interaction between a user and the system and would be somewhere in the middle of the abstraction scale. Not as abstract as a business process, and definitelly a lot more abstract than the implementation diagram.
Software projects tend to start working on a very abstract level (business goal for example) and finish with the abstracion 0 (implemented system). During the project analysts, architects and developers work together to gradually lower this abstraction producing always less abstract artifacts/models - business processes, use cases, architecture, design, code.
After this introduction it is not hard to answer your question - any of those can be used first and that depends on type of your project and its size. SOme examples:
- A large project of development an ERP system. It is almost certain that in this kind of project the first thing to model is the business process. A long before even thinking of its functionality, the team must understand the business background. The best UML diagram for this is naturally the activity diagram. Some time after, when the process is clear and the high-level reqs known, the use case modelling can start.
- A middle sized of relativelly small project, with no complex processes in background (for example a mobile app development) can start directly with use cases, identifying the users and their features. Later on, these can be further refined using activities.
- A very small development of some interface, driver of communication gateway, highely technical, where even the user interaction is minimal, the modelling can start with the activities again, showing the concrete algorithm too be implemented. USe cases can be completelly skipped.
As a summary I would conclude that there are no unbreakable rules of this kind in software development. Each project is unique, each development methodology is unique, even each development team is special and unique. To think about "which diagram" to do first is straight and simply WRONG! Think about what kind of analysis or specification you need in a given moment - what is easiest and most usefull to be modelled. When this is clear - there are 13 UML diagrams to pick up from in order to optimally fulfill the aim.
Choice of UML diagram is the "HOW". More important than that is more often than not - the the "WHAT".