The general rules for firearms are set out in the Firearms Act. The RCMP has webpages that explain the differences in more details.
In Canada, a possession and acquisition licence (PAL) is generally required to, well, possess and acquire firearms.
To obtain the PAL for non-restricted firearms, a safety class must be taken. For restricted firearms, an additional course has to be taken.
The regulation of firearms is a shared jurisdiction between the federal government's power over criminal law and the provinces' powers over property and civil rights. As such, the licenses are issued by the provinces under the federal Act. Quebec imposes additional processes on acquisition of restricted firearms and a requirement to register non-restricted firearms (which no longer exist federally since 2011).
As the name suggest, the acquisition and possession of prohibited firearms are generally prohibited. Specific exceptions exist for e.g. grandfathering existing firearms before they were prohibited or certain old collector handguns. In general, obtaining a PAL for prohibited firearms or to import them (other than re-importation) is not possible. The legal basis for the general prohibition (and exceptions) can be found in section 12 of the Act:
12 (1) An individual who is otherwise eligible to hold a licence is not eligible to hold a licence authorizing the individual to possess prohibited firearms except as provided in this section. [...]
The RCMP page on Classes of firearms also has more information. Ownership of restricted firearms must have a legal purpose (e.g. target shooting, collecting). Self-defence is not a legal purpose unless one is in an occupation with particular needs (e.g. armored truck guards) or a specific threat exist (under very, very, very, very limited circumstances). The purpose must be proved to a certain extent. For example, for target shooting, a membership in a club is required (although it is not difficult at all to obtain one).
The storage, display and transportation of firearms by individuals are regulated by the Storage, Display, Transportation and Handling of Firearms by Individuals Regulations under the Act.
The regulations are in relatively plain language. In short, few restrictions exist for non-restricted firearms (as long as they are unloaded, for storage and display, they must also be rendered inoperable by a secure means, e.g. locked). For restricted firearms, another layer of security is usually required (e.g. in an additional locked secure container).
For restricted firearms, an authorizations to transport is needed from the provincial chief firearms officer. However, since 2015, PAL holders "no longer have to apply separately in order to transport those firearms to certain routine activities such as target shooting; taking a firearm home after a transfer; going to a gunsmith, gun show, a Canadian port of exit or a peace officer or a CFO for verification, registration or disposal".
Note that carrying a weapon without a lawful excuse is a criminal offence in Canada under section 89 of the Criminal Code. Concealed carry of a weapon without a specific authorization is always a criminal offence under section 90. Potential self-defence is generally not a lawful excuse under Canadian law.
Provinces and municipalities may also have laws, ordinances, bylaws over the discharge of firearms. The federal regulations incorporate these laws, the contraventions of which may be prosecuted under subsection 86(2) of the Criminal Code.
The discharge of non-restricted firearms is not federally regulated as long as it is not towards another person, dangerous, reckless or otherwise offensive to the public peace.
The use of restricted firearms must be consistent with the purpose for which you possess such firearms. If your purpose is target shooting, you may only do so at an authorized shooting range. The CFO may also issue an authorization to transport with specific conditions authorizing certain uses.