If we buy several back links for the home page of the main domain, will it also help the sub-domains and sub-pages of our site?
Buying backlinks is risky. Google has stated that manual actions can be taken against your site for doing so.
If you receive backlinks to your domain, your subdomains won't necessarily benefit from it because it might be treated as individual domains depending upon how Google associates your subdomain to your maindomain. The stronger the association between your subdomain and domain, the more link juice the backlink is likely to carry to your subdomain.
If your main domain links to your subdomains it will pass link juice. So whatever backlinks you have pointing to your main domain, those can pass some juice to your subdomains as well.