The IoM tourism board is generally very helpful - look at www.gov.im - the site hosting this board also has many advertisers in hotels etc - choose according to your budget.
Where you want to stay depends on your interests - with a 14 year old I suspect you will not be spending much of your valuable time in the Manx Museum !
The public transport system on the Island is generally excellent - you can get short period combined season tickets for the buses/steam railway & electric railway - Douglas is probably the optimum place to stay for ease of access to most places though the cheaper tourism accomodation (for package tours) is often in Port Erin in the south.
Cars can be readily hired though l suspect at a much higher cost than North Americans are used to - petrol (gas) is now over 90p ($1.70) a litre ! - but roads are narrow and distances short so a small car is all that is needed.
You can also obtain multisite Manx National Heritage tickets (Castle Rushen, Peel etc) - 3 to 4 days is just enough to allow you to visit most sites but at something of a rush.