Is a car the best way to see Switzerland and Bavaria?

The problem you have is that Bavaria and Neuschwanstein are in Germany (although Neuschwanstein is so close to the Austrian border the problem can be mitigated) but most of the Alps are in Austria or Switzerland. The reason this is a problem is because hiring a car in country A and dehiring in country B attracts high financial penalties; the car hire companies really dislike it.

Getting to Neuschwanstein is pretty easy by public transport as this page (http://www.neuschwanstein.de/englisch/tourist/howtoget.htm) on the Neuschwanstein castle homepage will show. Check the DB website (http://www.bahn.co.uk is the english language version) for deals for a family travelling.

Getting around the Alps is also perfectly doable via public transport although potentially expensive. http://www.myswitzerland.com has details of passes you can use that either include the major touristy attractions such as cable cars and certain trains, or else they offer % discounts for those attractions.

It really depends on your exact proposed itinerary, when, how long for and what your budget is. Conceivably you could buy rail passes (as a random example) for the entire trip from England through to Italy. http://www.seat61.com will give you info re trains throughout Europe (1st class is unnecessary, stick with 2nd) whilst the aforementioned DB site will give you a Europe wide train timetable.

PS: Noting you're a yank, you should appreciate that fuel prices are the equivalent of circa $9 USD per US gallon and that automatics are not that common. Specify you want an automatic car in advance to the hire car company to ensure you get one, diesel would give you better mileage and shield you a bit from the motoring costs. Tolls for motorways apply in some areas (e.g. you need a sticker in Switzerland) - refer to the excellent site http://www.theaa.com/motoring_advice/overseas/index.html which will provide all the info you need.

You could e.g. travel from Paris via Strasbourg to Germany. Rent a car in Offenburg or Freiburg, travel through Switzerland and Bavaria and return the car there. Thus avoiding the often pretty steep international drop off fees.

Public transport in Switzerland and also Bavaria is excellent.
The SwissTravelSystem has a good deal for children under 16 years. Switzerland will be - whatever you do - however a rather expensive destination.
A good deal for Bavaria is the Bayern Ticket (Bavaria Ticket). EUR 29 for up to 5 persons traveling together. A day ticket covering all regional trains in Bavaria. Plus nearly all other means of local public transport. Munich - Füssen (for Neuschwanstein Castle) is anyway served by regional trains only.

Every place in Switzerland can be reached by public transport, but some mountain villages have no road access: Wengen, Muerren, Riederalp, Bettmeralp, Zermatt, Braunwald can only be reached by railway resp. cableway. Most trains and buses run every 30 to 60 minutes between 6am and midnight. Swiss trains are expensive, but things get cheaper if you buy a half fare card for 110 CHF/120 USD/90 EUR. Adults pay then only half the fare and kids pay nothing, if you ask for a free Family Card.
http://www.swisstravelsystem.ch/en/

Tickets for the direct Paris - Berne TGV can be got for 40 to 80 EUR/pax, if you book them in advance in the net. Some tickets for the Paris - Geneva TGV are sold for 25 EUR.
http://www.tgv-lyria.com/main/FCK/File/site_en/home/home_france_suisse.asp

Neuschwanstein is 3 hrs by German train and bus (Fare: 33 EUR) from Lindau a small town on an Island in Lake Constance which can be reached from Switzerland by train or boat.
http://www.swisstravelsystem.ch/fileadmin/pdf/STS-GB-M-11-EN-WEB.pdf

For travels in (German) Bavaria, check the Laender ticket:
http://www.bahn.de/i/view/USA/en/prices/germany/laender-ticket.shtml

General info about Switzerland: http://www.myswitzerland.com/en/home.html (open the links and the links of the links....)

Excellently covered across all angles already - here.
If I could just put in my bob's worth....
Rent-a-car would be the way forward for me. Sure, fuel is expensive - but will probably work out cheaper than four public transport tickets to anywhere and back.
The cops in Switzerland and Germany show no mercy for Speeding drivers. The penalties are very harsh - and dished out on-the-spot.
Don't speed - and you'll be just fine.
Go for the car ! !

For Switzerland: If there are no kids younger than 16 years in this "Family of 4" a car may be cheaper, but only if they focus on mountain places accessible by car, like Grand Saint Bernard, Stelvio, Verbier, Thyon 2000, Sanetschpass....
It would be a complete nonsense to rent a car and to visit places like Jungfraujoch, Titlis, Zermatt, etc., to do day long boat rides along Swiss waterways (like Geneva - Chillon Castle - Bouveret Steam Park, Yverdon - Murten, Murten - Solothurn, Schaffhausen - Rorschach) or to plan a Glacier or Bernina Express ride.