While it may seem obvious for team members to move on to the next highest items in the product backlog, I would advise against starting with this.
First and foremost, the teams' obligation is to achieving the sprint goal, so anything they can do to work towards that must come first (e.g. helping out testing, chipping in where possible, etc.).
Next, the team should look at expanding their definition of "done". Perhaps it currently doesn't include testing, or doesn't include some form of code review. Most teams starting with Scrum do not start with a definition of done that truly has a product increment that is ready to ship, so now would be the team to move towards that.
As others have mentioned, what tools do you need setup in order to get closer to a shippable state? Continuous integration? Automated acceptance tests? Now is the time to add these things.
Likely, you also have areas of the code that existed before you moved to Scrum and thus don't have very good test coverage or have accumulated technical debt. Now's the time to pay that off.
Also, as Mike Cohn suggests in his book Succeeding with Agile teams may want to reserve roughly 10% of their time for some look ahead planning. This may involve having a meeting with the Product Owner to discuss some up and coming stories for future sprints, breaking down larger stories into smaller ones, or for designers, perhaps doing some wire frames or mock-ups for upcoming stories.
Once you've gotten to this state, only then should you consider continuing on with the product backlog.