Akershus is historically divided into the traditional districts of Follo and Romerike, which make up most of the county, with the exception of the enclave west of Oslo, consisting of Asker and Bærum, which was created after Aker municipality was transferred to Oslo in 1948.
Rental cars and taxis are often available, but are rather expensive in Norway in general. The public transportation system is relatively good with buses going between the major cities and stopping at bus stops marked with a sign (blue with a white bus on it, rectangular) and/or an open air booth with the place name written above with white lettering. Public transport is reasonably inexpensive though costs can build up if you buy tickets every time you get on, day/week/month/season passes can be purchased at the major stops. At the central bus station in any town, cashiers will speak English.
Climb "Kolsåstoppen", a hill in Bærum. Nice view over the Oslo Fjord fram the top. There's also an even better hill in Asker with an even more excellent view.
Norway has high water quality and tap water is completely safe. If you still want bottled water, most stores carry several brands of non-artifically treated water.
Most areas in Akershus are more or less crime free, and you should not be afraid to walk around alone day or night, though you should always take precationary measures and follow your common sense when in a new country. Signs and maps are usually in Norwegian, but do not be afraid to ask for directions; the vast majority of Norwegians speak enough English to help out a lost tourist.