Area to stay in Florence

Florence is very safe everywhere, so you don't have to worry about safety.
The hotel you mention is next to Ponte Vecchio, therefore very central.
To determine the most central area for sight visiting draw a 1km radius circle with the center in Piazza della Repubblica. Nearly all sights in Florence will be inside that circle, therefore any hotel in that area will work.
http://www.freemaptools.com/radius-around-point.htm
enter Piazza Della Repubblica, Florence in the search box
Type 1 in the Radius Distance (km box)
Click on the map on piazza Della Repubblica to draw a circle with 1km radius
Click on zoom to fit map in the Map Options
Et Voila. That's your target area for your hotel searches.

We stayed last spring at Hotel La Scaletta, across the Arno from the hotel you mentioned and just a few blocks from Ponte Vecchio. We had a delightful room with a view of Boboli Gardens. The staff was so helpful and cheery. The website has a chat room with the front desk staff and a Facebook page. We chatted with Sarah, Sylvia, and Monica, all who helped us with Italian phrases. They also told us where THEY took their families to eat, get gelato, and enjoy views. The area was within easy walking distance to the major sights. Tell them George and Elaine mentioned their hotel. We still correspond with the front desk ladies on Facebook.

I like the area around the Duomo, I recently stayed at Hotel Bigallo just around the corner.

Thank you for the information! I will be arriving via train from Venice so I will not have a car. I would prefer to walk to my hotel from the train station but a taxi would be okay too.

If you stay near the Ponte Vecchio a walk from the station is possible but if you have a lot of luggage it might be better to take a taxi. The cost would not be excessive, probably less than 15 euro.

I checked out the Hotel La Scaletta's website. The prices are very reasonable and the rooftop terrace looks amazing! I really enjoy having a balcony or terrace to use. I will also check out the Duomo area and Hotel Bigallo. Thanks for the great recommendations!

between the duomo and the central market is also an easy walk from train station.....many food options....

Thanks again for all of the suggestions! I keep going back to the Ponte Vecchio area but it sounds like everyone is relatively close so perhaps it doesn't matter.

I recommend la Scaletta also. Roof bar and restaurant with Billion (with a B) dollar view.
Besides the Oltrarno (south side of the river) is slightly less overwhelmed by tourists and has retained some of the traditional "feel" of Florence of yesteryear when I was a kid. The Duomo area is Disneyland on steroids.

The old center of Florence, with all the tourist sights, is about 1/2 hour walk from one end to the other and the train station on the edge, so to speak. I would not recommend walking with luggage to your hotel. Cobblestone streets, crowded areas, and the possibility of a wrong turn, all make it easier to take a short taxi ride.

Waiting in line for the taxi at the station will probably take as much time as a walk to your hotel, but if you have lots of luggage, a taxi will be best. My RS rolling carryon has survived cobblestones all over Italy for several years, I've seen people with the 4-wheel luggage having problems (wheels are small and get stuck).

Thank you so much for the great suggestions! I like to walk to my hotel from the train station if it is within 1/2 mile or so but I don't mind a taxi either. Thanks for the good reviews on La Scaletta. I always love a good terrace to enjoy a cold drink at the end of a sightseeing day! Thanks again!!