Changing hosting doesn't affects your SEO Significantly but here are some precautions you may take a note of
1. Buy hosting from a good company- chose a professional hosting company. If possible, go for a dedicated IP or even for a VPS, if you budget allows you. Perform a blacklist check for the new IP (use mxtoolbox). Here I have compared some top VPS companies- Amir Nasir's answer to What are the best cheap VPS hosts?-Quora.
2. Location of the server- It affects the load time of your website. Make sure your server location is nearby the targeted audience location. (If you are targeting global audience then go for a US based server). If you're in India, take a look at this answer Which are the top VPS hosting providers in India? - Quora
3. Make sure, the hosting transfer has been done properly. Downtime should be minimum during transfer.
Again, It doesn't makes a major change to your SEO but one must do it correctly.