For the most part, there should be no change in rankings when changing hosting companies. We are assuming that your domain name is staying the same and that only your IP address is changing. Changing your IP address for your hosted website in itself should not affect the actual rankings.
DNS TTL (time to live)
Set your DNS TTL (time to live) very short, to around 5 minutes. This will help with the search engine spiders finding your new IP address.
You are going to have a copy of your website on two different host as you will be changing the DNS with your domain registrar. When you make the DNS change, you should see the DNS servers update within the time you specified. Keeping your site live on both hosts is recommended. Generally you will have a few weeks, or a month remaining on the old host. Keep the sites live on both, with the same exact pages/copy on each host. For your emails, to make sure there are no missed emails, you can opt to forward them to another email account (for your old hosting company). You will want to make sure you go with a good reliable hosting company, and that they have good support and very little downtime.
Not all hosting companies are equal, and when you find a good hosting company, you will want to stay with them as long as possible.
When changing hosting, you will want to make sure there is enough bandwidth with your hosting package.
Take note of all actual downtime with your website. If your host is having too many issues with downtime. See about having them switch your site to another server, or if that is not an option, and the downtime continues, you may need to look at changing hosting again.
Technical Hosting Support. This is a very important part and for myself a major consideration of changing hosting if the host has little hosting support. There most likely will always be issues that come up with hosting from time to time, and if your hosting technical leaves it up to you to solve it, changing hosting may be the answer.
Slow Responses to your problems. Another possible problem with hosting, is very slow responses to your emails.
Both technical support and slow responses were two major issues that resulted in the changing of hosting companies for my own websites.
Emails and SoapSpam
Not directly related to rankings, but you will always want to keep a watch on your email if your email is on a shared IP, which is normally the case. There have been times where SoapSpam have blocked an IP due to spam coming from one or multiple sites. For the most part, you will not need to worry about this, but keep tabs on it.
So the actual changing of an IP in itself should not affect your SEO, but there are other considerations that you will need to keep track of to make sure you are completely happy with your hosting. When switching, ask friends, co-workers, check the whois on some top websites to see where they may be hosted.