> Can I set a rule that says that if a user has never visited the page before he will see activity 1?
Yes, there is a "new visitor" attribute you can use to create an audience to apply to "activity 1" that limits it only to new visitors
> After the visitor sees activity 1 three times, then he will see activity 2 going forward?
Yes - use profile scripts to keep track
You can use the priorities to determine which experience will be shown to a visitor in the case that a visitor qualifies for 2 different Activities.
Here's the documentation - https://experienceleague.adobe.com/docs/target/using/activities/priority.html?lang=en
Consider this carefully when designing your testing and personalization strategy.
1. Avoid having multiple activities with same success metric or conversions on the same page - you'll cannibalize the traffic and confuse your results. Therefore, run them synchronously as much as possible.
2. If you must run multiple activities on the same page, you can use profile scripts / attributes to randomly place people into different "buckets" so that they do not overlap.
Definitely nuanced and particular to your testing/personalization strategy, so seek further advice from your consulting expert!