My husband and I are thinking of traveling at the beginning of August to Denmark and Sweden. I thought that we would possibly stay 5 nights in Copenhagen and 5 nights in Stockholm and travel around using those 2 cities as our home base. Any thoughts if this is a good idea or too much time in each location?

You'll probably want at least 2-3 days in each city... maybe more depending on your interests.
There are lots of lovely sites to visit outside the cities. Will you rent a car? (Altho in Stockholm you can take boats to many places.)
Also... August is a good month to travel in Scandinavia.

What is your recommendation for getting around? Also, any thoughts as to the best way to get from Copenhagen to Stockhom? Thanks for the info.

We flew on SAS from Stockholm to Copenhagen; it was less than an hour and less expensive than the train.
Five days in Stockholm would allow you to see most of the sights and take a boat ride out to the archipeligo too.
We loved Stockholm. We spent a week in Stockholm and three days in Copenhagen.

Perfect time to be there and that will give yo several days in each city plus time for a day trip. I wold look at flights between cities - then you need no other transit than feet (or a quick train or boat for day trip).