changing host

[idr]

I manage a database site with about 2500 items approximately 1.5 Gb volume and 600 to 1500 hits daily. After 7 years of hosting, my hosting company doesn’t give me a satisfactory support service, and also doubled the hosting fee. Lately Ii made a renewal for one more year but I am considering changing the hosting service.
All the necessary data are saved on my computer and I can easily upload them to another server. However I am afraid of interruptions in the service when transferring the site.
What will be the procedure of transfer, and what happens if the data are available simultaneously on both servers with the same domain name?
please advise
thanks