I'm planning to spend 6-7 days in Croatia. Is there a central place to stay to be able to see the sights or do I need to move from place to place
I didn't mean a central spot to see the entire country. I meant a spot for six to seven days
For 6 or 7 days, I don't think you need to move from hotel to hotel. If you choose to change hotels more than once you'll really cut into your sightseeing time. You really can't see both northern and southern Croatia in that amount of time no matter how often you switch hotels.
But before you get too heavily into developing your target-destination list, you need to consider how you're going to enter and leave the country. Links to Italy and Slovenia are less frequent than you might expect. On such a short trip you probably don't want to spend an entire day reaching your first (or only) hotel. On the other hand, you might turn up a viable open-jaw situation that could work for you with just a few hours on a bus (something like into Dubrovnik and out of Split, for example). It wouldn't be a disaster to change hotels once on a 6- or 7-day trip.
A spot near the northern part of the coast would allow access to the many picturesque towns on the Istrian Peninsula (interior as well as coastal, and Pula has some Roman ruins), the Skocjan and/or Postojna Caves, the northern islands (including Cres and Krk), and possibly the Plitvice Lakes National Park (though you'd either want to drive and get a really early start or spend the night before your visit very near the park, because of crowding). Rijeka and Opatija would be possible base cities if you prefer a central location rather than one of the picturesque but more isolated Istrian towns.
Along the south-central coast you have the Zadar-Split stretch, with access to several islands, including Brac and Hvar. Both of those cities are very nice, but six or seven days might be a bit more than you'd need on this stretch unless you're looking for beach time.
The far southern coast around Dubrovnik brings Mostar (Bosnia) and Kotor (Montenegro) into play. Other Montenegro destinations would also be possible. And there are island options as well.
Your range would be much greater with a car than if you are dependent on public transportation. If you'd be tied to buses, the BusCroatia website will be helpful. For a combination of buses, ships and the rare train, Rome2Rio is a possible starting point.
I would stay in Dubrovnik and see the surrounding islands by boat and poss up to Split for a day.