Here is the checklist for switching hosting provider:
Few more remarks about website configuring in new server. If you haven't made your website by yourself, you may need there some help. You may contact either with company which made your website or ask web hosting staff to do configurations for you. Most good web hosting providers offer moving your website to their server free of extra charge.
Changing of domain name server data will not take effect immediately. Usually it takes about 4 hours when changes are propogated around the net and domain will be associated with your new hosting server. If you have access to domain nameservers configuration, you may change them by yourself. Just find new server name servers names, they are usually in format ns1.domain.com and ns2.domain.com and associate them with your website domain in the domain management console.
In case you need to find better web hosting provider than your current one, we have some suggestions on web hosting top providers page.