I would like to travel to Rome in mid-September for one week with my family. Hence I am thinking of buying a weekly public transport pass. However I would like to know the range I can travel with this pass. For example if I want to visit the sea for one or two days, is this weekly pass valid?
The weekly public transport pass in Rome (in Italian CIS = Carta Integrata Settimanale) is valid within the area called "Roma Capitale".
You can buy it for € 24,00 at any automatic vending machine located in every metro station or at almost any newsagent in Rome. With the weekly pass you can ride all the buses, trams, metros and trains within the validity area. It will expire at 24:00 of the 7th day, including the day (no matter the time) of the first use.
The complete area coverage for metros and railways can be seen in this map. The red stops indicate the last stop in the lines within the "Roma Capitale" area, where the ticket can be used. As you can see, you can get to the sea by the so called Roma–Lido railway, which is fully included in the coverage area. If you want to reach other places by the see you will have to buy additional tickets depending on where you want to go. Note that none of the two airports are covered by the weekly ticket.
Alternatively, there are also regional tickets which cover extended zones, but they are much more expensive. You can find all the info at http://www.atac.roma.it. (Unfortunately only very few info are in English, but Google Translate could be helpful.)