Hey, since you are going in Feb, staying anywhere else away from the old town does not make sense as all the "action" is in the old town or right outside old town, specifically in these winter periods where there are less tourists, less things open etc.

For tips and tricks for the old town and what to do and where to stay inside of it I would suggest checking out this guide: Dubrovnik Old Town: Guide for Hotels, Things to do, Restaurants, Hidden Gems, Practical Tips

Now if you want to save some money and stay a bit outside of old town, walking distance 20-30 minutes I would recommend Lapad, plenty of hotels there so you might also find one with heated pool, I remember Rixos was has really nice spa included with the hotel.

To best decide on where to stay for cheaper check this guide: Where to stay when visiting Dubrovnik (8 Amazing areas) but since its Feb. I would recommend in or very near Old Town, during the summer I would advise opposite just due to amount of tourists etc.

Getting around Dubrovnik is really easy with Uber/Bolt or Busses: How to get around Dubrovnik

Finally you most likely will be visiting the city walls and I would recommend this personally done it more than 5 times, it gives you unique view over the city, great for pics. And get Dubrovnik pass as you pay the same amount as only the walls entry ticket but also get entrance to plenty of museums which is great for Feb cold weather.

Dubrovnik Pass: How to save money when visiting Dubrovnik (Guide)

If you will have any other questions just ask!