You need to follow certain step-by-step guide to achieve a painless and hassle free movement to your new affordable web site hosting provider.

1. Firstly, you need to download all current files used in your web site including graphics, html files etc that may be used in your site. Also make sure when you are copying your files you may want to save the fold structure exactly as it is on your original website server. Ideally, it is better to plan the transfer process at least one month before you terminate your old web hosting plan, this will give you sufficient time to perform a smooth transfer.
2. What kind of web site do you have, basic html or with many of particular features? You should to think your site requires, before choose a host that can supply what you demand. Nowdays many hosting company allow you to either do the domain change yourself or otherwise. 
3. While you are waiting for your new bluehost web hosting account to be ready, you need to take a backup all your web files on the old web hosting provider. Make sure you backup everything and anything you can.
4. Before you make changes to your domain name, make sure you load up all the files to your new host such as hostgator using temporary login information. Most hosting company do offer temp login account before your domain resolving to the new IP. Even though they won't really work yet, you can still go ahead and set up all the email accounts that are used on your domain.
5. No matter what registrar you used to register your web site domain, you can start the name change. Technically the only changes in your DNS record is the name server detail.
6. Depending on your domain registrar, you need to get one or more emails confirming the domain name change. Once the change has been initiated it usually takes around 24 to 48 hours for you and your users to see the change. This time is normally called propagation. All the DNS servers around the globe take the same amount of time to "catch up" and take note of your domains new location.