My wife and I are considering a trip to Portugal in late summer, 2013. We are taking our 2 children, 13 and 16. Although we will want to see a few sights, this is to be primarily a beach holiday.

I understand that the Algarve area is very desirable for this type of holiday. Can anyone recommend accommodations for us? Our budget is up to $300-$350 per night. A hotel on the beach would be preferable, but I don't know if that is possible, or which hotels or what specific area to start with. Also, we may travel with another family, with children, so a villa would also work.

Any ideas on where to start looking for a suitable place to stay, as well as things to do while staying in this area?

Your questions raises questions!
Portugal uses the Euro, besides which dollar are you talking about? Do you want an all inclusive? Do you want a large hotel or a small B&B type of establishment?
For hotels look at www.booking.com

Check out Praia da Rocha or Portimao as possible resorts, both have a range of hotels for most budgets as well as aparthotels. There are waterparks nearby for the kids and lots of restaurants and fast food places.

Most villas won't be on the seafront and you will pay a premium for any that are. For a choice of villas, you can have a look at:

http://www.ownersabroad.org/portugal.html
or
http://www.ownersdirect.co.uk/portugal.htm

Maybe Praia da Rocha but not Portimao - I would go for either Albufeira (old town not the sleazy new part) or Lagos and then consider the hotels there.

nothing wrong with Portimao, best hotels are on the cliff to the west or across the river in the villages to the east