The basic idea of moving a website from one server/host to another, cannot be associated with any ranking changes despite changing the IP address of the website. However, you could experience a temporary rank drop if there is downtime during the transfer. If you are transferring over to UltaHost, our expert technical support team will help you plan the transfer and minimize or completely remove downtime from the equation by using the best practices in the industry.
The only other way your SEO can be affected is if the new host is very fast compared to the previous one. You might see your ranking rising as search engines prioritize websites that load in 2 seconds or less. Note that speed is only one of the ~200 different factors that play a role in your ranking, so the impact will not be that significant.
The third big concern is running a website on a shared hosting environment. Google has confirmed that not having a Dedicated IP, which is very common if you are hosted on a shared hosting plan will not negatively impact your SEO, despite other websites with different page ranks using the same IP address.