You have to check with the potential web hosts. Many have storage and bandwidth limits and typically the starter packages start out around 500 to 750mb and that includes ALL of the stuff for your site, the wordpress installation, all databases and any uploads, images, themes, plugins, etc. Here at wordpress.COM, the 3GB limit is just for uploads. You never have to worry about how many posts you have, etc. Also know that when a host advertises "unlimited" bandwidth and storage, that isn't really true in most cases and if you start using a lot of bandwidth, they will throttle you back and then your site becomes very slow and you lose visitors, and if you start using huge amounts of storage they can come back and ask you to pay more money. I just had to separate one clients account into two separate accounts and split the sites up between them since the web host had flipped out due to the storage requirements and shut them down.
There are also a lot of responsibilities that come with self-hosting. You have to do the installation, all upgrades, all backups and all troubleshooting. If something goes wrong when you install a plugin or a theme and it kills your blog (this happens way more often than you would imagine), you have to figure it out and fix it, and sometimes that requires manually editing the database tables. Those that do not have at least a moderate level of expertise working with, and understanding of, web hosting and how it works can sometimes end up with troubles they have to hire someone to fix.
You should figure hosting will cost you $10 to $25 per month (stay completely away from "cheap" and "free" hosting as you get what you pay for.
You can point the domain you bought here at the new web host and use it with the new installation from your domain management section of your dashboard here.
One last note: Be very careful who your sources are for themes and plugins and such as there are some gutter dwellers out there that are taking popular free themes and plugins and redistributing them with malware in them.